Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation
LLarge deviation principle for the cutsets and lower large deviationprinciple for the maximal flow in first passage percolation ∗Barbara Dembin † , Marie Théret ‡ Abstract:
We consider the standard first passage percolation model in the rescaled lattice Z d /n for d ≥ R d . We denote by Γ and Γ two disjoint subsets of ∂ Ω representingrespectively the sources and the sinks, i.e. , where the water can enter in Ω and escape from Ω. A cutsetis a set of edges that separates Γ from Γ in Ω, it has a capacity given by the sum of the capacities ofits edges. Under some assumptions on Ω and the distribution of the capacities of the edges, we alreadyknow a law of large numbers for the sequence of minimal cutsets ( E minn ) n ≥ : the sequence ( E minn ) n ≥ converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutsetin an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce bycontraction principle a lower large deviation principle for the maximal flow in Ω. We use here the same notations as in [4]. Let n ≥ Z dn , E dn )having for vertices Z dn = Z d /n and for edges E dn , the set of pairs of points of Z dn at Euclidean distance1 /n from each other. With each edge e ∈ E dn we associate a capacity t ( e ), which is a random variablewith value in R + . The family ( t ( e )) e ∈ E dn is independent and identically distributed with a common law G . We interpret this capacity as a rate of flow, i.e. , it corresponds to the maximal amount of water thatcan cross the edge per second. Throughout the paper, we work with a distribution G on R + satisfyingthe following hypothesis. Hypothesis 1.
There exists
M > such that G ([ M, + ∞ [) = 0 and G ( { } ) < − p c ( d ) . Here p c ( d ) denotes the critical parameter of Bernoulli bond percolation on Z d .Let Ω be a bounded domain in R d . Let Γ and Γ be two disjoint subsets of the boundary ∂ Ω of Ωthat represent respectively the sources and the sinks. We aim to study the minimal cutsets that separateΓ from Γ in Ω for the capacities ( t ( e )) e ∈ E dn . We shall define discretized versions for those sets. For x = ( x , . . . , x d ) ∈ R d , we define k x k = vuut d X i =1 x i , k x k = d X i =1 | x i | and k x k ∞ = max (cid:8) | x i | , i = 1 , . . . , d (cid:9) . ∗ Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016) and the Labex MME-DII (ANR11-LBX-0023-01). † ETH Zürich. [email protected] ‡ Modal’X, UPL, Univ Paris Nanterre, F92000 Nanterre France and FP2M, CNRS FR [email protected] a r X i v : . [ m a t h . P R ] F e b e use the subscript n to emphasize the dependence on the lattice ( Z dn , E dn ). Let Ω n , Γ n , Γ n and Γ n bethe respective discretized version of Ω, Γ, Γ and Γ :Ω n = (cid:26) x ∈ Z dn : d ∞ ( x, Ω) < n (cid:27) , Γ n = (cid:8) x ∈ Ω n : ∃ y / ∈ Ω n , h x, y i ∈ E dn (cid:9) , Γ in = (cid:26) x ∈ Γ n : d ∞ ( x, Γ i ) < n , d ∞ ( x, Γ − i ) ≥ n (cid:27) , for i = 1 , , where d ∞ is the L ∞ distance associated with the norm k · k ∞ and h x, y i represents the edge whoseendpoints are x and y . We denote by Π n the set of edges that have both endpoints in Ω n , i.e. ,Π n = (cid:8) e = h x, y i ∈ E dn : x, y ∈ Ω n (cid:9) . Throughout the paper, Ω , Γ , Γ satisfy the following hypothesis: Hypothesis 2.
The set Ω is an open bounded connected subset of R d , it is a Lipschitz domain andits boundary Γ = ∂ Ω is included in a finite number of oriented hypersurfaces of class C that intersecteach other transversally. The sets Γ and Γ are two disjoint subsets of Γ that are open in Γ such that inf {k x − y k , x ∈ Γ , y ∈ Γ } > , and their relative boundaries ∂ Γ Γ and ∂ Γ Γ have null H d − measure,where H d − denotes the ( d − -dimensional Hausdorff measure. (Γ n , Γ n ) -cutset in Ω n . A set of edges E n ⊂ Π n is a (Γ n , Γ n )-cutset in Ω n if for any path γ from Γ n to Γ n in Ω n , γ ∩ E n = ∅ . We denote by C n (Γ , Γ , Ω) the set of (Γ n , Γ n )-cutset in Ω n . For E n ∈ C n (Γ , Γ , Ω),we define its capacity V ( E n ) and its associated measure µ n ( E n ) by V ( E n ) = X e ∈E n t ( e )and µ n ( E n ) = 1 n d − X e ∈E n t ( e ) δ c ( e ) (1.1)where c ( e ) denotes the center of the edge e and δ c ( e ) the dirac mass at c ( e ). The set E n is a discrete set,but in the limit it is more convenient to work with a continuous set. We first define r( E n ) ⊂ Z dn byr( E n ) = (cid:8) x ∈ Z dn : there exists a path from x to Γ n in ( Z dn , Π n \ E n ) (cid:9) . We define upon r( E n ) a continuous version R( E n ) by settingR( E n ) = r( E n ) + 12 n [ − , d . Hence, we have R( E n ) ∩ Z dn = r( E n ). Minimal cutsets.
A set E n ∈ C n (Γ , Γ , Ω) is a minimal cutset in Ω n if we have V ( E n ) = inf (cid:8) V ( F n ) : F n ∈ C n (Γ , Γ , Ω) (cid:9) . We denote by C n (0) the set of minimal cutsets in Ω n . We will often use the notation E minn to denote anelement of C n (0) chosen according to a deterministic rule. Minimal cutsets are the analogous in dimension d − d − Almost minimal cutsets.
Let ε >
0. A set E n ∈ C n (Γ , Γ , Ω) is a (Γ n , Γ n ) ε -cutset in Ω n if for any F n ∈ C n (Γ , Γ , Ω), we have V ( E n ) ≤ V ( F n ) + εn d − . We denote by C n ( ε ) the set of (Γ n , Γ n ) ε -cutset in Ω n . Note that the typical size of an almost minimalcutset is of surface order, that is of order n d − (see for instance lemma 3.2).2 aximal flow. We define φ n the maximal flow between Γ n and Γ n in Ω n as follows φ n (Γ , Γ , Ω) = inf (cid:8) V ( E n ) : E n ∈ C n (Γ , Γ , Ω) (cid:9) . The reason why this quantity is called a maximal flow is due to the max-flow min-cut theorem that statesthat the study of the minimal capacity of a cutset is the dual problem of the study of the maximal flow.Just as the study of minimal capacity is linked with the study of cutsets, the study of maximal flow islinked with the study of streams (a stream is a function that describes a stationary circulation of waterin the lattice). We won’t define rigorously what a stream is and its link with maximal flow. We refer forinstance to the companion paper [5] where we study large deviation principle for admissible streams toobtain an upper large deviation principle for the maximal flow.
We want to define the possible limiting objects for R( E n ) and µ n ( E n ) where E n ∈ C n ( ε ). Continuous cutsets.
We denote by C < ∞ the set of subsets of Ω having finite perimeter in Ω, i.e. , C < ∞ = (cid:8) E Borelian subset of R d : E ⊂ Ω , P ( E, Ω) < ∞ (cid:9) . When E is regular enough, its perimeter in the open set Ω corresponds to H d − ( ∂E ∩ Ω). We will givea more rigorous definition of the perimeter later (see (1.7)). Let E ∈ C < ∞ . We want to build from ∂E a continuous surface that would be a continuous cutset between Γ and Γ (we don’t give a formaldefinition of what a continuous cutset is). However, a continuous path from Γ to Γ does not have tointersect ∂E in general. For regular sets E and Ω, such a path should intersect b E = ( ∂E ∩ Ω) ∪ (Γ \ ∂E ) ∪ ( ∂E ∩ Γ )We define E as a more regular version of b E (see figure 1), E = ( ∂ ∗ E ∩ Ω) ∪ ( ∂ ∗ Ω ∩ ((Γ \ ∂ ∗ E ) ∪ ( ∂ ∗ E ∩ Γ )) , (1.2)where X denotes the closure of the set X and ∂ ∗ is the reduced boundary, we will give a rigorousdefinition later (see section 1.2.2).Figure 1 – Representation of the set E (the dotted lines) for some E ∈ C < ∞ .Note that in order to obtain a lower large deviation principle for the minimal cutsets, it is not enoughto keep track of the localization of the cutset. Indeed, if we only know the localization of the discreteminimal cutset it does not give information on its capacity. We will not only need the total capacity of acutset but also its local distribution. For a given localization of the minimal cutset and a given capacity,there are different macroscopic configurations where there exists a discrete cutset at the given localizationwith the given capacity. However this different macroscopic configurations do not have necessarily thesame cost, i.e. , the same probability to be observed. This is why we need to know locally the capacity.In the continuous setting, this boils down to introducing a function f : E → R + . The local capacity at3 point x in E will be given by f ( x ). Naturally, the total capacity will be obtained by summing thesecontributions. We define Cap( E, f ) as the capacity of the cutset E equiped with a local capacity f ( x )at x : Cap( E, f ) = Z E f ( x ) d H d − ( x ) . The smaller f ( x ) is the bigger the cost of having this local capacity is. We will see that there is a localcapacity that is in some sense costless. For x ∈ E and n ( x ) the associated normal exterior unit vector(of E or Ω depending whether x ∈ ∂ ∗ E or x ∈ ∂ ∗ Ω), the typical local capacity is ν G ( n ( x )) where ν G is acalled the flow constant. In other words, the probability that the local capacity is close to ν G ( n ( x )) at x is almost 1.We say that ( E, f ) is minimal if for any F ⊂ Ω of finite perimeter we haveCap(
E, f ) ≤ Z F ∩ E f ( y ) d H d − ( y ) + Z ( F \ E ) ∩ ∂ Ω ν G ( n Ω ( y )) d H d − ( y ) + Z ( F \ E ) \ ∂ Ω ν G ( n F ( y )) d H d − ( y )where F is defined for F in (1.2). This condition will be useful in what follows. This condition is verynatural, if we start with a minimal cutset that is close to E and with local capacity f , we expect that thelimiting object inherits this property. On E the local capacity is f but everywhere else the local capacityis the typical one given by the flow constant ν G . The right hand side may be interpreted as the capacityof F in the environment where the local capacity on E is f . Let T be the following set: T = ( E, f ) ∈ C < ∞ × L ∞ ( E → R , H d − ) : ( E, f ) is minimal, Cap(
E, f ) ≤ d M H d − (Γ ) ,f ( x ) ≤ ν G ( n E ( x )) H d − -a.e. on ∂ ∗ E ∩ Ω ,f ( x ) ≤ ν G ( n Ω ( x )) H d − -a.e. on ∂ ∗ Ω . We also define T M as follow T M = (cid:8) ( E, f H d − | E ) : ( E, f ) ∈ T (cid:9) . For two Borelian sets E and F , we define d ( E, F ) = L d ( E ∆ F ) where ∆ denotes the symmetric differenceand L d is the d -dimensional Lebesgue measure. Remark 1.1.
Let E n ∈ C n (Γ , Γ , Ω) . If lim n →∞ d (R( E n ) , E ) = 0 for some E ∈ B ( R d ) and µ n ( E n ) weakly converges towards µ , then we do not necessarily have that µ is absolutely continuous with respect to H d − | E . More generally, for any sequence ( E n ) n ≥ of Borelian sets of R d such that lim n →∞ d ( E n , E ) = 0 ,we do not have necessarily that H d − | ∂E n weakly converges towards H d − | ∂E . However, we can provethat if instead of studying any sequence of cutsets we study a sequence of minimal cutsets, then in thelimit µ will be supported on E . But, since it is too difficult to ensure we build a configuration of thecapacities of the edges with a minimal cutset at a given localization (it is difficult to ensure that thecutset we have built is indeed minimal), we will work instead with almost minimal cutsets, i.e., ε -cutsets.They are more flexible than minimal cutsets and their continuous limit is in the set T M . We denote by S d − the unit sphere in R d . Let ( E, f ) ∈ T . For any v ∈ S d − , we denote by J v the ratefunction associated with the lower large deviation principle for the maximal flow in a cylinder orientedin the direction v (see theorem 1.13). We can interpret J v ( λ ) as the cost of having a local capacity λ which is abnormally small in the direction v ( λ < ν G ( v )). It will be properly defined later in theorem1.13. We define the following rate function: I ( E, f ) = Z ∂ ∗ E ∩ Ω J n E ( x ) ( f ( x )) d H d − ( x ) + Z ∂ ∗ Ω ∩ ((Γ \ ∂ ∗ E ) ∪ (Γ ∩ ∂ ∗ E )) J n Ω ( x ) ( f ( x )) d H d − ( x ) . Roughly speaking I ( E, f ) is the total cost of having a cutset E with the local capacities given by f , theoverall cost is equal to the sum of the contributions of the local costs over the continuous cutset E .We denote by M ( R d ) the set of positive measures on R d . We endow M ( R d ) with the weak topology O . We denote by B the σ -field generated by O . We endow the set B ( R d ) of Borelian sets of R d with thetopology O associated with the distance d . We denote by B the σ -field associated with this distance.Let n ≥ ε > P εn denotes the following probability: ∀ A ∈ B ⊗ B P εn ( A ) = P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ A ) .
4e define the following rate function e I on B ( R d ) × M ( R d ) as follows: ∀ ( E, ν ) ∈ B ( R d ) × M ( R d ) e I ( E, ν ) = (cid:26) + ∞ if ( E, ν ) / ∈ T M I ( E, f ) if ν = f H d − | E with ( E, f ) ∈ T . The following theorem is the main result of this paper.
Theorem 1.2 (Large deviation principle on cutsets) . Let G that satisfies hypothesis 1. Let (Ω , Γ , Γ ) that satisfies hypothesis 2. The sequence ( P εn ) n ≥ satisfies a large deviation principle with speed n d − governed by the good rate function e I and with respect to the topology O ⊗ O in the following sense: forall A ∈ B ⊗ B− inf (cid:8)e I ( ν ) : ν ∈ ˚ A (cid:9) ≤ lim ε → lim inf n →∞ n d − log P εn ( A ) ≤ lim ε → lim sup n →∞ n d − log P εn ( A ) ≤ − inf (cid:8)e I ( ν ) : ν ∈ A (cid:9) . Remark 1.3.
Because of the limit in ε , it is not a proper large deviation principle. Unfortunately, wewere not able to obtain a large deviation principle on E minn because of the lower bound (see remark 1.1).However, this result is enough to prove a lower large deviation principle on the maximal flow. We can deduce from theorem 1.2, by a contraction principle, the existence of a rate function governingthe lower large deviations of φ n (Γ , Γ , Ω). Let J be the following function defined on R + : ∀ λ ≥ J ( λ ) = inf ne I ( E, ν ) : (
E, ν ) ∈ B ( R d ) × M ( R d ) , ν ( R d ) = λ o and define λ min as λ min = inf { λ ≥ J ( λ ) < ∞} . The real number φ Ω > φ n (Γ , Γ , Ω) /n d − . Wehave the following lower large deviation principle for the maximal flow. Theorem 1.4 (Lower large deviation principle for the maximal flow) . Let G that satisfies hypothesis1. Let (Ω , Γ , Γ ) that satisfies hypothesis 2. The sequence ( φ n (Γ , Γ , Ω) /n d − , n ∈ N ) satisfies a largedeviation principle of speed n d − governed by the good rate function J .Moreover, the map J is finite on ] λ min , φ Ω ] , infinite on [0 , λ min [ ∪ ] φ Ω , + ∞ [ , and we have ∀ λ < λ min ∃ N ≥ ∀ n ≥ N P ( φ n (Γ , Γ , Ω) ≤ λn d − ) = 0 . (1.3) Remark 1.5.
The property (1.3) is used to prove upper large deviation principle for the maximal flowin the companion paper [5]. It also gives the precise role of λ min in our study. In the companion paper [5], we proved the upper large deviation principle for the maximal flow.
Theorem 1.6 (Upper large deviation principle for the maximal flow) . Let G that satisfies hypothesis1. Let (Ω , Γ , Γ ) that satisfies hypothesis 2. The sequence ( φ n (Γ , Γ , Ω) /n d − , n ∈ N ) satisfies a largedeviation principle of speed n d with the good rate function e J u . Moreover, there exists λ max > such that the map e J u is convex on R + , infinite on [0 , λ min [ ∪ ] λ max , + ∞ [ , e J u is null on [ λ min , φ Ω ] and strictly positive on ] φ Ω , + ∞ [ . We refer to [5] for a precise definition of e J u . Theorems 1.4 and 1.6 give the full picture of largedeviations of φ n (Γ , Γ , Ω). The lower large deviations are of surface order since it is enough to decreasethe capacities of the edges along a surface to obtain a lower large deviations event. The upper largedeviations are of volume order, to create an upper large deviations event, we need to increase thecapacities of constant fraction of the edges. This is the reason why to study lower large deviations,it is natural to study cutsets that are ( d − d -dimensional objects. Actually, theorem 1.4 is used in [5] to prove theorem 1.6. Theorem 1.4justifies the fact that the lower large deviations are not of the same order as the order of the upper largedeviations. 5 .2 Background We now present the mathematical background needed in what follows. We present a maximal flowin cylinders with good subadditive properties and give a rigorous definition of the limiting objects.
Let A be a non-degenerate hyperrectangle, i.e. , a rectangle of dimension d − R d . Let v ∈ S d − such that v is not contained in an hyperplane parallel to A . Let h >
0. If v is one of the two unit vectorsnormal to A , we denote by cyl( A, h ) the following cylinder of height 2 h :cyl( A, h ) = (cid:8) x + tv : x ∈ A, t ∈ [ − h, h ] (cid:9) . We have to define discretized versions of the bottom half B ( A, h ) and the top half T ( A, h ) of theboundary of the cylinder cyl(
A, h ), that is if we denote by z the center of A : T ( A, h ) = (cid:8) x ∈ Z dn ∩ cyl( A, h ) : ( z − x ) · v > ∃ y / ∈ cyl( A, h ) , h x, y i ∈ E dn (cid:9) , (1.4) B ( A, h ) = (cid:8) x ∈ Z dn ∩ cyl( A, h ) : ( z − x ) · v < ∃ y / ∈ cyl( A, h ) , h x, y i ∈ E dn (cid:9) , (1.5)where · denotes the standard scalar product in R d . We denote by τ n ( A, h ) the maximal flow from theupper half part to the lower half part of the boundary of the cylinder, i.e. , τ n ( A, h ) = inf { V ( E ) : E cuts T ( A, h ) from B ( A, h ) in cyl(
A, h ) } . The random variable τ n ( A, h ) has good subadditivity properties since minimal cutsets in adjacent cylin-ders can be glued together along the common side of these cylinders by adding a negligible amount ofedges. Therefore, by applying ergodic subadditive theorems in the multi-parameter case, we can obtainthe convergence of τ n ( A, h ) /n d − . Theorem 1.7 (Rossignol-Théret [12]) . Let G be a measure on R + such that G ( { } ) < − p c ( d ) . For any v ∈ S d − , there exists a constant ν G ( v ) > such that for any non-degenerate hyperrectangle A normalto v , for any h > , we have lim n →∞ τ n ( A, h ) H d − ( A ) n d − = ν G ( v ) a.s. . The function ν G is called the flow constant. This is the analogue of the time constant defined upon thegeodesics in the standard first passage percolation model where the random variables ( t ( e )) e representpassage times. Remark 1.8. If G ( { } ) ≥ − p c ( d ) , the flow constant is null (see Zhang [14]). It follows that φ Ω = 0 and there is no interest of studying lower large deviations. We present here some result on upper large deviations for the random variable τ n ( A, h ) that will beuseful in what follows. It states that the upper large deviation for the random variable τ n ( A, h ) are ofvolume order.
Theorem 1.9 (Théret [13]) . Let v be a unit vector, A be an hyperrectangle orthogonal to v and h > .Let us assume that G satisfies hypothesis 1. For every λ > ν G ( v ) , we have lim inf n →∞ − H d − ( A ) n d log P (cid:18) τ n ( A, h ) H d − ( A ) n d − ≥ λ (cid:19) > . Let us first recall some mathematical definitions. For a subset X of R d , we denote by X the closureof X , by ˚ X the interior of X . Let a ∈ R d , the set a + X corresponds to the following subset of R d a + X = { a + x : x ∈ X } . r >
0, the r -neighborhood V i ( X, r ) of X for the distance d i , that can be Euclidean distance if i = 2or the L ∞ -distance if i = ∞ , is defined by V i ( X, r ) = (cid:8) y ∈ R d : d i ( y, X ) < r (cid:9) . We denote by B ( x, r ) the closed ball centered at x ∈ R d of radius r >
0. Let v ∈ S d − . We define theupper half ball B + ( x, r, v ) and the lower half ball B − ( x, r, v ) as follows: B + ( x, r, v ) = { y ∈ B ( x, r ) : ( y − x ) · v ≥ } and B − ( x, r, v ) = { y ∈ B ( x, r ) : ( y − x ) · v < } . We also define the disc disc( x, r, v ) centered at x of radius r and of normal vector v as follows:disc( x, r, v ) = { y ∈ B ( x, r ) : ( y − x ) · v = 0 } . For n ≥
1, we define the discrete interior upper boundary ∂ + n B ( x, r, v ) and the discrete interior lowerboundary ∂ − n B ( x, r, v ) as follows: ∂ + n B ( x, r, v ) = (cid:26) y ∈ B ( x, r ) ∩ Z dn : ∃ z ∈ Z dn \ B ( x, r ) , ( z − x ) · v ≥ k z − y k = 1 n (cid:27) and ∂ − n B ( x, r, v ) = (cid:26) y ∈ B ( x, r ) ∩ Z dn : ∃ z ∈ Z dn \ B ( x, r ) , ( z − x ) · v < k z − y k = 1 n (cid:27) . For U ⊂ Z dn , we define its edge boundary ∂ e U by ∂ e U = (cid:8) h x, y i ∈ E dn : x ∈ U, y / ∈ U (cid:9) . (1.6)For F ∈ B ( R d ) and δ >
0, we denote by B d ( F, δ ) the closed ball centered at F or radius δ for the topologyassociated with the distance d , i.e. , B d ( F, δ ) = (cid:8) E ∈ B ( R d ) : L d ( F ∆ E ) ≤ δ (cid:9) . Let C b ( R d , R ) be the set of continuous bounded functions from R d to R . We denote by C kc ( A, B ) for A ⊂ R p and B ⊂ R q , the set of functions of class C k defined on R p , that takes values in B and whosedomain is included in a compact subset of A . The set of functions of bounded variations in Ω, denotedby BV (Ω), is the set of all functions u ∈ L (Ω → R , L d ) such thatsup (cid:26)Z Ω div h d L d : h ∈ C ∞ c (Ω , R d ) , ∀ x ∈ Ω h ( x ) ∈ B (0 , (cid:27) < ∞ . Rectifiability and Minkowski content.
We will need the following proposition that enables to relatethe measure of a neighborhood of a set E with its Hausdorff measure. Let p ≥
1. Let M be a set suchthat H p ( M ) < ∞ . We say that a set M is p -rectifiable if there exists countably many Lipschitz maps f i : R p → R d such that H p M \ [ i ∈ N f i ( R p ) ! = 0 . Proposition 1.10.
Let p ≥ . Let M be a subset of R d that is p -rectifiable. Then we have lim r → L d ( V ( M, r )) α d − p r d − p = H p ( M ) where α d − p denote the volume of the unit ball in R d − p . In particular, for p = d − , we have lim r → L d ( V ( M, r ))2 r = H d − ( M ) . d − Sets of finite perimeter and surface energy.
The perimeter of a Borel set E of R d in an open setΩ is defined as P ( E, Ω) = sup (cid:26)Z E div f ( x ) d L d ( x ) : f ∈ C ∞ c (Ω , B (0 , (cid:27) , (1.7)where C ∞ c (Ω , B (0 , C ∞ from R d to B (0 ,
1) having a compact supportincluded in Ω, and div is the usual divergence operator. The perimeter P ( E ) of E is defined as P ( E, R d ).The topological boundary of E is denoted by ∂E . The reduced boundary.
For E a set of finite perimeter, we denote by χ E its characteristic function.The distributional derivative ∇ χ E of χ E is a vector Radon measure and P ( E, Ω) = k∇ χ E k (Ω) where k∇ χ E k is the total variation measure of ∇ χ E . The reduced boundary ∂ ∗ E of E is a subset of ∂E suchthat, at each point x of ∂ ∗ E , it is possible to define a normal vector n E ( x ) to E in a measure-theoreticsense, that is points such that ∀ r > k∇ χ E k ( B ( x, r )) > r → − ∇ χ E ( B ( x, r )) k∇ χ E k ( B ( x, r )) = n E ( x ) . For a point x ∈ ∂ ∗ E , we havelim r → r d L d (( E ∩ B ( x, r ))∆ B − ( x, r, n E ( x ))) = 0 , (1.8)and for H d − almost every x in ∂ ∗ E ,lim r → α d − r d − H d − ( ∂ ∗ E ∩ B ( x, r )) = 1 (1.9)where α d − is the volume of the unit ball in R d − . Moreover, we have ∀ A ∈ B ( R d ) k∇ χ E k ( A ) = H d − ( ∂ ∗ E ∩ A ) . By De Giorgi’s structure theorem (see for instance theorem 15.9 in [8]), the reduced boundary is ( d − β >
0. We define the set C β as the sets of Borelian subsets of Ω of perimeter less than β C β = n F ∈ B ( R d ) : F ⊂ Ω , P ( F, Ω) ≤ β o (1.10)endowed with the topology associated to the distance d . For this topology, the set C β is compact.The following little lemma will appear several times in what follows. We refer to [1] for a proof ofthis lemma. Lemma 1.11 (Lemma 6.7 in [1]) . Let f , . . . , f r be r non-negative functions defined on ]0 , . Then, lim sup ε → ε log r X i =1 f i ( ε ) ! = max ≤ i ≤ r lim sup ε → ε log f i ( ε ) . We work here with the same environment as in section 1.1.1. For any Borelian set F ⊂ Ω such that P ( F, Ω) < ∞ , we define its capacity I Ω ( F ) as follows I Ω ( F ) = Z Ω ∩ ∂ ∗ F ν G ( n F ( x )) d H d − ( x ) + Z Γ ∩ ∂ ∗ F ν G ( n F ( x )) d H d − ( x ) + Z Γ ∩ ∂ ∗ (Ω \ F ) ν G ( n Ω ( x )) d H d − ( x ) . x ∈ ∂ ∗ F in the direction n F ( x ) in the lattice ( Z dn , E dn ) is ν G ( n F ( x )). The capacity I Ω ( F ) correspondsto Cap( F, f ) where f ( x ) = ν G ( n F ( x )) for x ∈ ∂ ∗ F and f ( x ) = ν G ( n Ω ( x )) for x ∈ Γ ∩ ∂ ∗ (Ω \ F ). Wecan interpret the capacity I Ω ( F ) as the capacity of the continuous cutset F . We can prove that, almostsurely, there exist cutsets in C n (Γ , Γ , Ω) localized in some sense close to F and of capacity of order I Ω ( F ) n d − . Denote by φ Ω the minimal continuous capacity, i.e. , φ Ω = inf {I Ω ( F ) : F ⊂ Ω , P ( F, Ω) < ∞} . (1.11)Let us denote by Σ a the set of continuous cutsets that achieves φ Ω Σ a = { F ⊂ Ω : P ( F, Ω) < ∞ , I Ω ( F ) = φ Ω } . In [4], Cerf and Théret proved a law of large numbers for the maximal flow and the minimal cutsetin the domain Ω. The minimal cutset converges in some sense towards Σ a . Theorem 1.12. [Cerf-Théret [4]] Let G that satisfies hypothesis 1 and such that G ( { } ) < − p c ( d ) (toensure that ν G is a norm). Let (Ω , Γ , Γ ) that satisfies hypothesis 2. Then the sequence ( E minn ) n ≥ (werecall that E minn ∈ C n (0) ) converges almost surely for the distance d towards the set Σ a , that is, a.s., lim n →∞ inf F ∈ Σ a d (R( E minn ) , F ) = 0 . Moreover, we have lim n →∞ φ n (Γ , Γ , Ω) n d − = φ Ω > . They first prove that from each subsequence of (R( E minn )) n ≥ they can extract a subsequence thatconverges for the distance d towards a set F ⊂ Ω such that P ( F, Ω) < ∞ . Using locally the law of largenumbers for the maximal flow in a cylinder (theorem 1.7), they prove thatlim inf n →∞ V ( E minn ) n d − ≥ I Ω ( F ) ≥ φ Ω . The remaining part of the proof required working with maximal streams (which is the dual objectassociated with minimal cutset). We refer to the companion paper [5] for a more precise study ofmaximal streams.
In [10], Rossignol and Théret proved a lower largedeviation principle for the variable τ . The rate function they obtain is going to be our basic brick tobuild the rate function for lower large deviation principle in a general domain. Theorem 1.13 (Large deviation principle for τ ) . Suppose that G ( { } ) < − p c ( d ) and that G has anexponential moment. Let h > . Then for every vector v ∈ S d − , for every non degenerate hyperrectangle A normal to v , the sequence (cid:18) τ n ( A, h ) H d − ( A ) n d − , n ∈ N (cid:19) satisfies a large deviation principle of speed H d − ( A ) n d − governed by the good rate function J v . More-over, we know that that J v is convex on R + , infinite on [0 , δ G k v k [ ∪ ] ν G ( v ) , + ∞ [ where δ G = inf { t, P ( t ( e ) ≤ t ) > } , equal to at ν G ( v ) , and if δ G k v k < ν G ( v ) , we also know that J v is finite on ] δ G k v k , ν G ( v )] ,continuous and strictly decreasing on [ δ G k v k , ν G ( v )] and strictly positive on ] δ G k v k , ν G ( v )[ . Lower large deviations for the maximal flow in a domain.
We here work with the same environ-ment as in section 1.1.1. Cerf and Théret proved in [2] that the lower large deviations for the maximalflow φ n through Ω are of surface order. 9 heorem 1.14. If (Ω , Γ , Γ ) satisfy hypothesis 2 and the law G of the capacity admits an exponentialmoment, i.e., there exists θ > such that Z R + exp( θx ) dG ( x ) < ∞ and if G ( { } ) < − p c ( d ) , then there exists a finite constant φ Ω > such that ∀ λ < φ Ω lim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) < . Remark 1.15.
Note that this constant φ Ω is the same than in (1.11) . To prove this result, on the lower large deviation event, they consider the continuous subset E n =R( E minn ) where E minn is a minimal cutset E minn . Since we can control the number of edges in a minimalcutset thanks to the work of Zhang [15], we can control the perimeter of E n in Ω and work with highprobability with a continuous subset E n with perimeter less than some constant β >
0. The set E n belongs to the compact set C β , this enables to localize the set E n close to some set F ∈ C β . To conclude,the idea is to say that since the capacity of E minn is strictly smaller that I Ω ( F ) n d − , then there existslocally a region on the boundary of F where the flow is abnormally low. We can relate this event withlower large deviations for the maximal flow in a small cylinder, whose probability decay speed is ofsurface order. This large deviation result was used to prove the convergence of the rescaled maximalflow φ n /n d − towards φ Ω . This strategy was already using the study of a cutset but was too rough toderive a large deviation principle. Step 1. Admissible limiting object.
One of the main difficulty of this work was to identify the rightobject to work with. The aim is that the limiting objects must be measures supported on surfaces. Inparticular, we want to prove that the limiting object for (R( E n ) , µ n ( E n )) is contained in the set T M .Thanks to the compactness of the set C β , if we have a control on the perimeter of R( E n ) we can easilyprove that up to extraction, it converges towards a continuous Borelian subset E of Ω. If we work withgeneral cutsets without any restriction, the limiting object for µ n ( E n ) may not be a measure supportedon the surface E . This is due to the potential presence of long thin filaments for the set E n that are ofnegligible volume for the set R( E n ) but in the limit these filaments can create measure outside of E .Working with a minimal cutset E minn prevents the existence of these long filaments that are not optimalto minimize the capacity when G ( { } ) < − p c ( d ). This will ensure that the weak limit of µ n ( E minn ) issupported on E . However, working with minimal cutset leads to a major difficulty for the lower bound(see the next step). One solution is to work with almost minimal cutsets, i.e. , cutsets in C n ( ε ). We provethat in some sense the limiting objects for (( R ( E n ) , µ n ( E n )) , E n ∈ C n ( ε )) n ≥ are contained in the set T M . Step 2. Lower bound.
For any (
E, ν ) ∈ T M such that e I ( E, ν ) < ∞ , we prove that for any neighbor-hood U of ( E, ν ) the probability that there exists a cutset E n ∈ C n ( ε ) such that that ( R ( E n ) , µ n ( E n )) ∈ U is at least exp( − n d − e I ( E, ν )). Write ν = f H d − | E . To prove this result, we build a configuration wherethe expected event occurs using elementary events as building blocks. We first cover almost all the bound-ary of E by a finite family ( B ( x i , r i , v i )) i ∈ I of disjoint closed balls such that on each ball ∂E is almost flatand f is almost constant. Using result for lower large deviations for the maximal flow in cylinders, wecan prove that the probability that there exists in the ball B ( x, r, v ) a cutset separating ∂ − n B ( x, r, v ) from ∂ + n B ( x, r, v ) of capacity smaller than f ( x ) α d − r d − is of probability at least exp( − α d − r d − J v ( f ( x ))).Let us denote by E ( i ) this event associated with the ball B ( x i , r i , v i ) and E n ( i ) the cutset given in thedefinition of the event (if there are several possible choices, we pick one according to a deterministic rule).We can build a set of edges F of negligible cardinal such that E n := F ∪ ∪ i ∈ I E n ( i ) ∈ C n (Γ , Γ , Ω). Ifthe radii of the balls are small enough depending on the neighborhood U , we can prove that on the event10 i ∈ I E ( i ) , we have (R( E n ) , µ n ( E n )) ∈ U . Since the balls are disjoint, we have P ( ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ P \ i ∈ I E ( i ) ! = Y i ∈ I P ( E ( i ) ) ≥ exp − X i ∈ I J v i ( f ( x i )) α d − r d − i n d − ! ≈ exp (cid:16) − e I ( E, ν ) n d − (cid:17) . It remains to prove that E n is almost minimal, this is the main difficulty of this step. To do so, we haveto ensure that everywhere outside the balls, the flow is not abnormally low. This step is very technical.Using this strategy, we did not manage to prove that the cutset we have built is minimal but only almostminimal. Step 3. Upper bound.
In the lower bound section, we build a configuration upon elementary eventson balls. Here, we do the reverse. We deconstruct the configuration into a collection of elementaryevents on disjoint balls. Fix (
E, ν ) ∈ T M , write ν = f H d − | E . We first cover almost all the boundaryof E by a finite family ( B i = B ( x i , r i , v i )) i ∈ I of disjoint closed balls such that on each ball ∂E is almostflat and f is almost constant. We pick a neighborhood U of ( E, ν ) adapted to this covering such thatfor E n ∈ C n (Γ , Γ , Ω) such that (R( E n ) , µ n ( E n )) ∈ U , we have that for each i ∈ I the set E n ∩ B i is almost a cutset between ∂ − n B i and ∂ + n B i in B i (up to adding a negligible number of edges) and V ( E n ∩ B i ) ≤ f ( x i ) α d − r d − i . Hence, on the event {∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U} , theevent ∩ i ∈ I E ( i ) occurs where E ( i ) was defined in the previous step. We can prove using estimates on lowerlarge deviations on cylinders that P ( E ( i ) ) ≤ exp( − α d − r d − i J v i ( f ( x i )) n d − ). Since the balls are disjointit follows that P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ P \ i ∈ I E ( i ) ! = Y i ∈ I P ( E ( i ) ) ≤ exp − X i ∈ I α d − r d − i J v i ( f ( x i )) n d − ! ≈ exp( − e I ( E, ν ) n d − ) . The remaining of the proof uses standard tools of large deviations theory. The proof of this largedeviation principle for almost minimal cutsets enables to deduce by a contraction principle a lower largedeviation principle for the maximal flow.
Organization of the paper.
In section 2, we prove some useful results such as a covering theorem andlower large deviations for the maximal flow in a ball. In section 3 (corresponding to step 1), we studythe properties of the limiting objects to prove that they belong with high probability to a neighborhoodof the set T M . In section 4 (corresponding to step 2) and 5 (corresponding to step 3), we prove localestimates on the probability P εn ( U ) for some neighborhoods U to be able to deduce a large deviationprinciples for the almost minimal cutsets. Finally, in section 6, we conclude the proof of the two maintheorems 1.2 and 1.4. We will use the Vitali covering theorem for L d . A collection of sets U is called a Vitali class fora Borelian set Ω of R d , if for each x ∈ Ω and δ >
0, there exists a set U ∈ U such that x ∈ U and0 < diam U < δ where diam U is the diameter of the set U for the Euclidean distance. We now recallthe Vitali covering theorem for H d − (Theorem 1.10 in [6])11 heorem 2.1 (Vitali covering theorem) . Let F ⊂ R d such that H d − ( F ) < ∞ and U be a Vitali classof closed sets for F . Then we may select a countable disjoint sequence ( U i ) i ∈ I from U such thateither X i ∈ I (diam U i ) d − = + ∞ or H d − F \ [ i ∈ I U i ! = 0 . We next recall the Besicovitch differentiation theorem in R d (see for example theorem 13.4 in [1]): Theorem 2.2 (Besicovicth differentiation theorem) . Let µ be a finite positive Radon measure on R d .For any Borel function f ∈ L ( µ ) , the quotient µ ( B ( x, r )) Z B ( x,r ) f ( y ) dµ ( y ) converges µ -almost surely towards f ( x ) as r goes to . We recall that for any set E ∈ C < ∞ , the set E denotes the continuous cutset associated with E thatwas defined in (1.2). We will use at several moments in the proof the Vitali covering theorem. To avoidrepeating several times the same arguments, we here present a general result for covering a surface bydisjoint closed balls that satisfy a list of properties. These properties are typical for balls centered atpoints in the surface provided that their radius is small enough. Proposition 2.3 (Covering E by balls) . Let E ∈ C < ∞ . Let ε ∈ ]0 , / . Let ( P ) x,r , . . . , ( P m ) x,r be afamily of logical proposition depending on x ∈ R d , ε and r > . We assume that there exists R such that H d − ( E \ R ) = 0 and ∀ x ∈ R ∃ r x > such that ∀ i ∈ { , . . . , m } ∀ < r ≤ r x ( P i ) x,r holds . Then, there exists a finite family of disjoint closed balls ( B ( x i , r i , v i )) i ∈ I with v i = n Ω ( x i ) (respectively n E ( x i ) ) for x i ∈ ∂ ∗ Ω \ ∂ ∗ E (resp. x i ∈ ∂ ∗ E ) such that H d − ( E \ ∪ i ∈ I B ( x i , r i ))) ≤ ε , ∀ i ∈ I ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( E ∩ B ( x i , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , ∀ i ∈ I ∀ < r ≤ r i ∀ j ∈ { , . . . , m } ( P j ) x i ,r holds . At this point, the logical propositions ( P i ) x,r may seem a bit abstract. To better understand the kindof applications, we will do, let us give an example of such a proposition:( P ) x,r := (cid:28) x ∈ ∂ ∗ Ω = ⇒ L d ((Ω ∩ B ( x, r ))∆ B − ( x, r, n Ω ( x ))) ≤ εα d r d (cid:29) . Proof of proposition 2.3.
We follow the proof of lemma 14.6 in [1]. Let ε be a positive constant, with ε < / First case: x ∈ ∂ ∗ E . By inequality (1.9), we have for H d − -almost every x ∈ ∂ ∗ E lim r → α d − r d − H d − ( ∂ ∗ E ∩ B ( x, r )) = 1 . We denote by R the set of points in ∂ ∗ E such that the equality holds. Hence, we have H d − ( ∂ ∗ E \ R ) = 0 . (2.1)It yields that for x ∈ R , there exists a positive constant r ( x, ε ) > r ≤ r ( x, ε ) (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( ∂ ∗ E ∩ B ( x, r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . econd case: x ∈ ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E ) . By inequality (1.9), we have for H d − -almost every x ∈ ∂ ∗ Ωlim r → α d − r d − H d − ( ∂ Ω ∩ B ( x, r )) = 1 . We denote by R the set of points in ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E ) such that the previous equality holds. Hence, wehave H d − (cid:0) ( ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E )) \ R (cid:1) = 0 . (2.2)It yields that for x ∈ R , there exists a positive constant r ( x, ε ) ∈ ]0 , ε ] such that for any r ≤ r ( x, ε ) (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( ∂ ∗ Ω ∩ B ( x, r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . Extract a countable covering by balls.
The family of balls (cid:0) B ( x, r ) , x ∈ ( R ∪ R ) ∩ R , r < min(1 , r ( x, ε ) , r x ) (cid:1) is a Vitali class for ( R ∪ R ) ∩ R . By Vitali Covering theorem (theorem 2.1), we may select from thisfamily a countable (or finite) disjoint sequence of balls ( B ( x i , r i ) , i ∈ I ) such that either X i ∈ I r d − i = + ∞ or H d − (( R ∪ R ) ∩ R ) \ [ i ∈ I B ( x i , r i ) ! = 0 . We know that E and Ω both have finite perimeter. Since the balls in the family ( B ( x i , r i ) , i ∈ I ) aredisjoint, we have X i ∈ I H d − ( B ( x i , r i ) ∩ E ) ≤ H d − ( E ) < ∞ . We recall that for any i ∈ I , we have (cid:12)(cid:12) α d − r d − i − H d − ( B ( x i , r i ) ∩ E ) (cid:12)(cid:12) ≤ α d − r d − i ε . Hence, we have X i ∈ I r d − i ≤ α d − (1 − ε ) X i ∈ I H d − ( B ( x i , r i ) ∩ E ) < ∞ . As a result, we obtain that H d − ( R ∪ R ) ∩ R \ [ i ∈ I B ( x i , r i ) ! = 0 . (2.3)Consequently, we can extract from I a finite set I such that H d − ( R ∪ R ) ∩ R \ [ i ∈ I B ( x i , r i ) ! ≤ ε . We set v i = n Ω ( x i ) (respectively n E ( x i )) for x i ∈ ∂ ∗ Ω \ ∂ ∗ E (resp. x i ∈ ∂ ∗ E ). Since H d − ( E \ (( R ∪R ) ∩ R )) = 0, this concludes the proof. Ω inside balls centered at the boundary of Ω The following result will be important in what follows. Since the boundary of Ω is smooth, theintersection of Ω with a small ball B ( x, r ) centered at x ∈ Γ is close to B − ( x, r, n Ω ( x )). In the followinglemma, we localize precisely the symmetric difference between Ω ∩ B ( x, r ) and B − ( x, r, n Ω ( x )). Lemma 2.4.
Let Ω that satisfies hypothesis 1. Let x ∈ ∂ ∗ Ω . For any δ > , there exists r dependingon δ , x and Ω such that ∀ < r ≤ r (Ω ∩ B ( x, r ))∆ B − ( x, r, n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . roof. Let x ∈ ∂ ∗ Ω. We recall that Γ = ∂ Ω is contained in the union of a finite number of orientedhypersurfaces of class C that intersect each other transversally. We claim that if x ∈ ∂ ∗ Ω, it cannot becontained in the transversal intersection. Indeed, by definition for points in a transversal intersection,we cannot properly define an exterior unit normal vector at points in the intersection. It follows that Γis locally C around x . Let δ >
0. There exists r > ∀ < r ≤ r ∀ y ∈ B ( x, r ) ∩ Γ k n Ω ( y ) − n Ω ( x ) k ≤ δ . Since Ω is a Lipschitz domain, up to choosing a smaller r , there exists an hyperplane H containing x ofnormal vector n Ω ( x ) and φ : H → R a Lipschitz function such that B ( x, r ) ∩ Γ = { y + φ ( y ) n Ω ( x ) : y ∈ H ∩ B ( x, r ) } and B ( x, r ) ∩ Ω = { y + tn Ω ( x ) : y ∈ H ∩ B ( x, r ) , t ≤ φ ( y ) } ∩ B ( x, r )Let y ∈ B ( x, r ) ∩ ∂ Ω. Figure 2 – Reprentation of B ( x, r ), H , y and z .Let us denote by x and y the points in H such that x = x + φ ( x ) n Ω ( x ) and y = y + φ ( y ) n Ω ( x )(actually, x = x and φ ( x ) = 0, see figure 2). Let us denote by φ the following mapping ∀ s ∈ [0 , φ ( s ) = φ ((1 − s ) x + sy ) . Note that since ∂ Ω ∩ B ( x, r ) is contained in a C hypersurface, the mapping φ ∈ C ( R , R ). By themean value theorem, there exists s ∈ ]0 ,
1[ such that φ ( s ) = φ (1) − φ (0) = φ ( y ) − φ ( x ). In otherwords, the vector y − x + ( φ ( y ) − φ ( x )) n Ω ( x ) = y − x belongs to the tangent space of the point z = ((1 − s ) x + sy + φ ( s ) n Ω ( x )) ∈ Γ. Consequently, we have n Ω ( z ) · ( y − x ) = 0 . Hence, we get using Cauchy Schwartz inequality | ( y − x ) · n Ω ( x ) | = | ( y − x ) · ( n Ω ( z ) − n Ω ( x )) | ≤ k y − x k k n Ω ( z ) − n Ω ( x ) k ≤ δ k y − x k and ∂ Ω ∩ B ( x, r ) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . Let w ∈ Ω ∩ B + ( x, r, n Ω ( x )). There exists w ∈ H ∩ B ( x, r ) and 0 ≤ t ≤ φ ( w ) such that w = w + tn Ω ( x ).If φ ( w ) = 0, then we have w ∈ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . Let us assume φ ( w ) > w + φ ( w ) n Ω ( x ) ∈ Γ, we have φ ( w ) = | ( w + φ ( w ) n Ω ( x ) − x ) · n Ω ( x ) | ≤ δ k w + φ ( w ) n Ω ( x ) − x k .
14t follows that | ( w − x ) · n Ω ( x ) | = | ( w − x ) · n Ω ( x ) + t | = t = tφ ( w ) φ ( w ) ≤ tφ ( w ) δ k w + φ ( w ) n Ω ( x ) − x k ≤ δ tφ ( w ) k w − x k + δt ≤ δ ( k w − x k + t )= δ k w + tn Ω ( x ) − x k and w ∈ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . As a result, we getΩ ∩ B + ( x, r, n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . By the same arguments, we can prove thatΩ c ∩ B − ( x, r, n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } . The result follows.
Note that the starting point to understand large deviations for the maximal flow in general domainis to first understand large deviations in cylinders. Several times in this paper, we will cover surfacesby disjoint balls. Hence, we will need to understand how the maximal flow behaves in a ball. From ourknowledge on large deviations in cylinders, we can deduce results for large deviations in balls that is thebasic brick we need in this paper. We will need the following lemma that is an adaptation of what isdone in section 6 in [2]. Let n ≥
1. Let x ∈ R d , v ∈ S d − and r , δ , ζ be positive constants. We firstdefine G n ( x, r, v, δ, ζ ) to be the event that there exists a set U ⊂ B ( x, r ) ∩ Z dn such that:card( U ∆( B − ( x, r, v ) ∩ Z dn )) ≤ δα d r d n d and V (( ∂ e U ) ∩ B ( x, r )) ≤ ζα d − r d − n d − where we recall that ∂ e U denotes the edge boundary of U and was defined in (1.6). The set ( ∂ e U ) ∩ B ( x, r )correspond to the edges in ∂ e U that have both endpoints in B ( x, r ). Lemma 2.5.
There exists a constant κ depending only on d and M such that for any x ∈ R d , v ∈ S d − and r , δ , ζ positive constants, we have lim sup n →∞ n d − P ( G n ( x, r, v, δ, ζ )) ≤ − g ( δ ) α d − r d − J v ζ + κ √ δg ( δ ) ! where g ( δ ) = (1 − δ ) ( d − / .Proof of lemma 2.5. We aim to prove that on the event G n ( x, r, v, δ, ζ ), we can build a cutset thatseparates the upper half part of ∂B ( x, r, v ) (upper half part according to the direction v ) from the lowerhalf part that has a capacity close to ζα d − r d − n d − . To do so, we build from the set U an almost flatcutset in the ball. The fact that card( U ∆ B − ( x, r, v )) is small implies that ∂ e U is almost flat and is closeto disc( x, r, v ). However, this does not prevent the existence of long thin strands that might escape theball and prevent U from being a cutset in the ball. The idea is to cut these strands by adding edges at afixed height. We have to choose the appropriate height to ensure that the extra edges we needed to addto cut these strands are not too many, so that we can control their capacity. The new set of edges wecreate by adding to U these edges will be in a sense a cutset. The last thing to do is then to cover thedisc( x, r, v ) by hyperrectangles in order to use the rate function that controls the decay of the probabilityof having an abnormally low flow in a cylinder. Let ρ > δ . We define γ max = ρr . γ max will represent the height of a cylinder of basis disc( x, r , v ). We have to choose r insuch a way that cyl(disc( x, r , v ) , γ max ) ⊂ B ( x, r ). We set r = r p − ρ . On the event G n ( x, r, v, δ, ζ ), we consider a fixed set U satisfying the properties described in the definitionof the event. For each γ in { /n, . . . , ( b nγ max c − /n } , we define D ( γ ) = cyl(disc( x, r , v ) , γ ) ,∂ + n D ( γ ) = (cid:26) y ∈ D ( γ ) ∩ Z dn : ∃ z ∈ Z dn , ( z − x ) · v > γ and k z − y k = 1 n (cid:27) and ∂ − n D ( γ ) = (cid:26) y ∈ D ( γ ) ∩ Z dn : ∃ z ∈ Z dn , ( z − x ) · v < − γ and k z − y k = 1 n (cid:27) . The sets ∂ + n D ( γ ) ∪ ∂ − n D ( γ ) are pairwise disjoint for different γ . Moreover, we have X γ =1 /n,..., ( b nγ max c− /n card( U ∩ ∂ + n D ( γ ))+card( U c ∩ ∂ − n D ( γ )) ≤ card( U ∆( B − ( x, r, v ) ∩ Z dn )) ≤ δα d r d n d . By a pigeon-hole principle, there exists γ in { /n, . . . , ( b nγ max c − /n } such thatcard( U ∩ ∂ + n D ( γ )) + card( U c ∩ ∂ − n D ( γ )) ≤ δα d r d n d b nγ max c − ≤ δα d r d n d − γ max = 5 δα d r d − n d − ρ for n large enough. If there are several choices for γ , we pick the smallest one. We denote by X = U ∩ D ( γ ). We define by X + and X − the following set of edges: X + = {h y, z i ∈ E dn : y ∈ ∂ + n D ( γ ) ∩ X, z / ∈ D ( γ ) } ,X − = {h y, z i ∈ E dn : y ∈ ∂ − n D ( γ ) ∩ X c , z / ∈ D ( γ ) } . Let us control the number of edges in X + ∪ X − :card( X + ∪ X − ) ≤ d (cid:0) card( U ∩ ∂ + n D ( γ )) + card( U c ∩ ∂ − n D ( γ )) (cid:1) ≤ d δα d r d − n d − ρ = C d δρ − r d − n d − where C d = 10 dα d . We want to relate the event G n ( x, r, v, δ, ζ ) to flows in cylinders. There exists aconstant c d depending only on the dimension such that for any positive κ , there exists a finite collectionof closed disjoint hyperrectangles ( A i ) i ∈ J included in disc( x, r , v ) such that X i ∈ J H d − ( A i ) ≥ α d − r d − − κ and X i ∈ J H d − ( ∂A i ) ≤ c d r d − . (2.4)Since all the hyperrectangles are closed and disjoint, we have ξ = min { d ( A i , A j ) : i = j ∈ J } > . For any i ∈ J , we denote by P i ( n ) the edges with at least one endpoint in V (cyl( ∂A i , γ ) , d/n ). For n sufficiently large, we have ξ > d/n and all the sets P i ( n ) are pairwise disjoint. Moreover, usingproposition 1.10, we have X i ∈ J card( P i ( n )) ≤ d X i ∈ J L d ( V (cyl( ∂A i , γ ) , d/n )) n d ≤ X i ∈ J d H d − ( ∂A i )2 γ n d − ≤ d ρrc d r d − n d − ≤ d c d ρr d − n d − . We define E i = P i ( n ) ∪ (cid:0) ( X + ∪ X − ∪ ( ∂ e U ∩ D ( γ ))) ∩ cyl( A i , γ max ) (cid:1) .
16e can check that E i is a cutset for τ n ( A i , γ ) (we don’t prove here that the set E i is a cutset, we referto the proof of lemma 2.7 for a proof that a set is a cutset). The sets E i are pairwise disjoint and X i ∈ J V ( E i ) ≤ V ( ∂ e U ∩ B ( x, r )) + M X i ∈ J card( P i ( n )) + card( X + ∪ X − ) ! ≤ (cid:0) ζα d − + ( C d δρ − + 32 d c d ρ ) M (cid:1) r d − n d − . Set ρ = √ δ . We obtain that P ( G n ( x, r, v, δ, ζ )) ≤ P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i ⊂ E dn is a cutset in cyl( A i , γ max ) and P i ∈ J V ( E i ) ≤ ( ζ + κ √ δ ) α d − r d − n d − (cid:19) (2.5)where κ = M ( C d + 32 d c d ) /α d − . Let ε >
0. We want to sum on all possible values of (cid:18)(cid:24) V ( E i ) ε r d − n d − (cid:25) ε r d − n d − , i ∈ J (cid:19) . It is easy to check that X i ∈ J (cid:24) V ( E i ) ε r d − n d − (cid:25) ε r d − n d − ≤ X i ∈ J V ( E i ) + | J | ε r d − n d − ≤ (( ζ + κ √ δ ) α d − + | J | ε ) r d − n d − , where | J | denotes the cardinality of J . There are at most (( ζ + κ √ δ ) α d − /ε + | J | ) | J | possible valuesfor the family (cid:18)(cid:24) V ( E i ) ε r d − n d − (cid:25) , i ∈ J (cid:19) . Hence, the number of possible values is finite and does not depend on n . Let S be the following set S = ( ( β i ) i ∈ J ∈ N | J | : X i ∈ J β i ε r d − n d − ≤ (( ζ + κ √ δ ) α d − + | J | ε ) r d − n d − ) . By inequality (2.5), we obtain P ( G n ( x, r, v, δ, ζ )) ≤ P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i ⊂ E dn is a cutset in cyl( A i , γ max ) and P i ∈ J V ( E i ) ≤ ( ζ + κ √ δ ) α d − r d − n d − (cid:19) ≤ X ( β i ) i ∈ J ∈S P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i is a cutset for τ n ( A i , γ max ) and (cid:6) V ( E i ) /ε r d − n d − (cid:7) = β i (cid:19) . (2.6)Since the cylinders are all disjoint, using the independence, we have ∀ ( β i ) i ∈ J ∈ S P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i is a cutset for τ n ( A i , γ max ) and (cid:24) V ( E i ) ε r d − n d − (cid:25) = β i (cid:19) ≤ Y i ∈ J P (cid:0) τ n ( A i , γ max ) ≤ β i ε r d − n d − (cid:1) . Thanks to theorem 1.13, it follows thatlim sup n →∞ n d − log P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i is a cutset for τ n ( A i , γ max ) and (cid:24) V ( E i ) ε r d − n d − (cid:25) = β i (cid:19) ≤ X i ∈ J lim sup n →∞ n d − log P (cid:0) τ n ( A i , γ max ) ≤ β i ε r d − n d − (cid:1) ≤ − X i ∈ J H d − ( A i ) J v (cid:18) β i ε r d − H d − ( A i ) (cid:19) . J v it is a decreasing function, inequality (2.4) and the convexity of J v , we have for ( β i ) i ∈ J ∈ S , J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − α d − r d − − κ ! ≤ J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − H d − ( ∪ i ∈ J A i ) ! ≤ J v H d − ( ∪ i ∈ J A i ) X i ∈ J H d − ( A i ) (cid:18) β i ε r d − H d − ( A i ) (cid:19)! ≤ X i ∈ J H d − ( A i ) H d − ( ∪ i ∈ J A i ) J v (cid:18) β i ε r d − H d − ( A i ) (cid:19) . Combining the two previous inequalities and (2.4), we obtainlim sup n →∞ n d − log P (cid:18) ∃ ( E i ) i ∈ J : ∀ i ∈ J E i is a cutset for τ n ( A i , γ max ) and (cid:24) V ( E i ) ε r d − n d − (cid:25) = β i (cid:19) ≤ −H d − ( ∪ i ∈ J A i ) J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − α d − r d − − κ ! ≤ − ( α d − r d − − κ ) J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − α d − r d − − κ ! . Combining this inequality with inequality (2.6) and lemma 1.11 giveslim sup n →∞ n d − log P ( G n ( x, r, v, δ, ζ )) ≤ − ( α d − r d − − κ ) J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − α d − r d − − κ ! . Since J v is a good rate function, it is lower semi-continuous, hence,lim inf κ → lim inf ε → J v (( ζ + κ √ δ ) α d − + | J | ε ) r d − α d − r d − − κ ! ≥ J v ( ζ + κ √ δ ) α d − r d − α d − r d − ! . As a result, we obtainlim sup n →∞ n d − log P ( G n ( x, r, v, δ, ζ )) ≤ − α d − r d − (1 − δ ) ( d − / J v ζ + κ √ δ (1 − δ ) ( d − / ! where we use that r = r √ − δ . By setting g ( δ ) = (1 − δ ) ( d − / , the result follows. Let n ≥
1. Let x ∈ R d , v ∈ S d − and r , δ , ζ be positive constants. We first define G n ( x, r, v, δ, ζ ) asthe event that there exists a cutset E n in B ( x, r ) ∩ Ω n that cuts ∂ + n B ( x, r, v ) ∪ ((Γ n ∪ Γ n ) \ ∂ − n B ( x, r, v ))from ∂ − n B ( x, r, v ) such that E n ⊂ cyl(disc( x, r, v ) , δr )and V ( E n ) ≤ ζα d − r d − n d − . Lemma 2.6.
Let η > . Let δ > such that M d (1 − ( p − δ (1 − dδ )) d − + 4 δα d − ) ≤ η α d − . (2.7) • Let x ∈ Ω and r > such that B ( x, r ) ⊂ Ω . For any v ∈ S d − , we have lim sup n →∞ n d − log P (cid:0) G n ( x, r, v, , ν G ( v ) + η ) c (cid:1) = −∞ . For any ζ > , lim inf n →∞ n d − log P (cid:0) G n ( x, r, v, δ, ζ + η ) (cid:1) ≥ − α d − r d − J v ( ζ ) . Let x ∈ ∂ ∗ Ω ∩ (Γ ∪ Γ ) . Let r > such that (Ω ∩ B ( x, r, n Ω ( x )))∆ B − ( x, r, n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } , we have lim sup n →∞ n d − log P (cid:0) G n ( x, r, n Ω ( x ) , , ν G ( n Ω ( x )) + η ) c (cid:1) = −∞ . For any ζ > , lim inf n →∞ n d − log P (cid:0) G n ( x, r, n Ω ( x ) , δ, ζ + η ) (cid:1) ≥ − α d − r d − J n Ω ( x ) ( ζ ) . Proof of lemma 2.6.
Let η > δ we will choose later depending on η . First case: x ∈ Ω . Let v ∈ S d − and r >
0. We set h max = 2 δr and r = r p − δ . With such a choice of r , we have cyl(disc( x, r , v ) , h max ) ⊂ B ( x, r ). Let ( e , . . . , e d − , v ) be an or-thonormal basis. Let S δ be the hyper-square of side-length δr of normal vector v having for ex-pression [0 , δr [ d − ×{ } in the basis ( e , . . . , e d − , v ). We can pave disc( x, r , v ) with a family ( S i ) i ∈ I of translates of S δ such that the S i are pairwise disjoint, there are all included in disc( x, r , v ) anddisc( x, r (1 − dδ ) , v ) ⊂ disc( x, r − dδr, v ) ⊂ ∪ i ∈ I S i . We denote by E n ( i ) the cutset that achieves theinfimum in τ n ( S i , h max ). If there are several possible choices, we use a deterministic rule to break ties.Let F be the set of edges with at least one endpoint included in V (cid:0) disc( x, r, v ) \ disc( x, r (1 − dδ ) , v ) , d/n (cid:1) ∪ [ i ∈ I V ( ∂S i , d/n ) . Note that H d − ( ∂S δ ) = 2( d − δr ) d − where ∂S δ denotes the relative boundary, i.e. , the boundary of S δ in the hyperplane { x ∈ R d : x · v = 0 } . Using proposition 1.10, we have for n large enoughcard( F ) ≤ dn d L d V (disc( x, r, v ) \ disc( x, r (1 − dδ ) , v ) , d/n ) ∪ [ i ∈ I V ( ∂S i , d/n ) ! ≤ d α d − r d − (cid:18) − (cid:16)p − δ (1 − dδ ) (cid:17) d − (cid:19) n d − + 16 d | I | δ d − r d − α n d − ≤ d r d − (cid:18) − (cid:16)p − δ (1 − dδ ) (cid:17) d − (cid:19) n d − + 16 d r d − δ α α d − n d − . We choose δ small enough such that M d (cid:18) − (cid:16)p − δ (1 − dδ ) (cid:17) d − (cid:19) ≤ η α d − . Then, we choose n large enough such that16 d M r d − δ α α d − n d − ≤ η α d − r d − n d − . With this choice, we have M card( F ) ≤ η α d − r d − n d − . We notice that the set F ∪ ( ∪ i ∈ I E n ( i )) is a cutset that cuts ∂ + n B ( x, r, v ) from ∂ − n B ( x, r, v ) in B ( x, r ) ∩ Ω n . It follows that P (cid:0) G n ( x, r, v, , ν G ( v ) + η ) c (cid:1) ≤ P (cid:0) V ( F ∪ ( ∪ i ∈ I E n ( i ))) ≥ ( ν G ( v ) + η ) α d − r d − n d − (cid:1) ≤ P X i ∈ I τ n ( S i , h max ) ≥ | I | (cid:16) ν G ( v ) + η (cid:17) ( δr ) d − n d − ! ≤ X i ∈ I P (cid:16) τ n ( S i , h max ) ≥ (cid:16) ν G ( v ) + η (cid:17) ( δr ) d − n d − (cid:17) .
19y theorem 1.9, it follows thatlim sup n →∞ n d − log P (cid:0) G n ( x, r, v, , ν G ( v ) + η ) c (cid:1) = −∞ . Let ζ >
0. The set F n = F ∪ ( ∪ i ∈ I E n ( i )) is a cutset that cuts ∂ + n B ( x, r, v ) from ∂ − n B ( x, r, v ) in B ( x, r ) and is contained in cyl(disc( x, r, v ) , δr ). On the event ∩ i ∈ I { τ n ( S i , h max ) ≤ ζ ( δr ) d − n d − } , wehave V ( F n ) ≤ M card( F ) + X i ∈ I τ n ( S i , h max ) ≤ ηα d − r d − n d − + | I | ζ ( δr ) d − n d − ≤ ( ζ + η ) α d − r d − n d − and the event G n ( x, r, v, δ, ζ + η ) occurs. Besides, we have using the independence, P (cid:0) ∩ i ∈ I { τ n ( S i , h max ) ≤ ζ ( δr ) d − n d − } (cid:1) = Y i ∈ I P ( τ n ( S i , h max ) ≤ ζ ( δr ) d − n d − ) . It follows by theorem 1.13 thatlim inf n →∞ n d − log P ( G n ( x, r, v, δ, ζ + η )) ≥ X i ∈ I lim inf n →∞ n d − log P ( τ n ( S i , h max ) ≤ ζ ( δr ) d − n d − ) ≥ − α d − r d − ( δr ) d − ( δr ) d − J v ( ζ ) = − α d − r d − J v ( ζ ) . The result follows.
Second case: x ∈ ∂ ∗ Ω . We will only prove the case where x ∈ Γ since the proof for x ∈ Γ is similar.The proof is similar to the first case with extra technical difficulties. In particular, we cannot pavedisc( x, r, n Ω ( x )) directly because the cutset may exit Ω since B ( x, r ) Ω. To fix this issue, we are goingto move this cylinder slightly in the direction − n Ω( x ) . It is easy to check that { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δr }⊂ cyl(disc( x, r, n Ω ( x )) , δr ) . Set x = x − δrn Ω ( x ) . Using that { y ∈ B ( x, r ) : ( y − x ) · n Ω ( x ) ≤ − δr } ⊂ Ω (see figure 3), we havecyl(disc( x , r , n Ω ( x )) , δr/ ⊂ cyl(disc( x, r , n Ω ( x )) , δr ) ∩ Ω ⊂ B ( x, r ) ∩ Ω . x ∈ ∂ ∗ Ω.Let ( e , . . . , e d − , n Ω ( x )) be an orthonormal basis. Let S δ be the hyper-square of side-length δr ofnormal vector n Ω ( x ) having for expression [0 , δr [ d − ×{ } in the basis ( e , . . . , e d − , n Ω ( x )). We can pavedisc( x , r , n Ω ( x )) with a family ( S i ) i ∈ I of translates of S δ such that the S i are pairwise disjoint, there areall included in disc( x , r , n Ω ( x )) and disc( x , r (1 − dδ ) , n Ω ( x )) ⊂ ∪ i ∈ I S i . We denote by E n ( i ) the cutsetthat achieves the infimum in τ n ( S i , h max / F be the set of edges with at least one endpoint included in V (cid:0) disc( x , r, n Ω ( x )) \ disc( x , r (1 − dδ ) , n Ω ( x )) , d/n (cid:1) ∪ [ i ∈ I V ( ∂S i , d/n ) ! ∪ ( V ( ∂B ( x, r ) ∩ cyl(disc( x, r, n Ω ( x )) , δr ) , d/n )) . Note that H d − ( ∂B ( x, r ) ∩ cyl(disc( x, r, n Ω ( x )) , δr )) = 2 Z δr H d − (cid:0) ∂B ( x, r ) ∩ disc( x + u n Ω ( x ) , r, n Ω( x ) ) (cid:1) du ≤ δα d − r d − . By the same computations as in the previous case, we havecard( F ) ≤ d (cid:18) − (cid:16)p − δ (1 − dδ ) (cid:17) d − + 4 δα d − (cid:19) n d − + 16 d r d − δ α α d − n d − . We choose δ small enough such that M d (cid:18) − (cid:16)p − δ (1 − dδ ) (cid:17) d − + 4 δα d − (cid:19) ≤ η α d − . Then, we choose n large enough such that16 d M r d − δ α α d − n d − ≤ η α d − r d − n d − . With this choice M card( F ) ≤ η α d − r d − n d − . Set F n = F ∪ ( ∪ i ∈ I E n ( i )). The set F n may not be included in Π n , but we claim that F n ∩ Π n is acutset that cuts ∂ + n B ( x, r, n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, r, n Ω ( x ))) from ∂ − n B ( x, r, n Ω ( x )) in B ( x, r ) ∩ Ω n . Let γ be a path from ∂ + n B ( x, r, n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, r, n Ω ( x ))) to ∂ − n B ( x, r, n Ω ( x )) in Ω n ∩ B ( x, r ). Wewrite γ = ( x , e , x . . . , e p , x p ). If x ∈ Γ n , since Γ n ∩ B ( x, r ) ⊂ cyl(disc( x, r, n Ω ( x )) , δ ), then we have x · n Ω ( x ) ≥ − δr . If x ∈ ∂ + n B ( x, r, n Ω ( x )), then we have x · n Ω ( x ) ≥ Let us assume x p · n Ω ( x ) ≥ − δr . Since x p ∈ ∂ − n B ( x, r, n Ω ( x )), we have x p · n Ω ( x ) ≤
0. It followsthat x p ∈ V ( ∂B ( x, r ) ∩ cyl(disc( x, r, n Ω ( x )) , δr ) , d/n ) and h x p − , x p i ∈ F . . Let us assume x p · n Ω ( x ) < − δr . If γ ∩ V (cid:0) disc( x , r, n Ω ( x )) \ disc( x , r (1 − dδ ) , n Ω ( x )) , d/n (cid:1) ∪ S i ∈ I V ( ∂S i , d/n ) = ∅ then γ ∩ F = ∅ . Let us assume that γ ∩ V (cid:0) disc( x , r, n Ω ( x )) \ disc( x , r (1 − dδ ) , n Ω ( x )) , d/n (cid:1) ∪ S i ∈ I V ( ∂S i , d/n ) = ∅ . Then, there exists i ∈ I such that there exists an excursionof γ in cyl( S i , δr/
2) from the bottom half to the top half of the cylinder. It follows that γ ∩ E n ( i ) = ∅ .The set F n is indeed a cutset. We conclude similarly as in the first case.Lemma 2.6 can only be applied for radius that does not depend on n . We will work with balls withradius r > n . The following lemma enables to use the lemma 2.6 when the radius r depends on n but is close to some fixed r > Lemma 2.7.
Let < ε ≤ / . There exists κ d ≥ a constant depending only on d such that • For any δ > , for any x ∈ Ω and r > such that B ( x, r ) ⊂ Ω , for any v ∈ S d − , for n largeenough depending on r and ε , we have ∀ r ∈ (cid:2) (1 − √ ε ) r, r (cid:3) G n ( x, (1 − √ ε ) r, v, δ, ζ ) ⊂ G n ( x, r , v, δ, ζ + κ d M √ ε ) . • For any < δ ≤ ε , for any x ∈ ∂ ∗ Ω ∩ (Γ ∪ Γ ) , for any r > such that ∀ < r ≤ r (Ω ∩ B ( x, r , n Ω ( x )))∆ B − ( x, r , n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } (2.8) and ∀ < r ≤ r (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( ∂ ∗ Ω ∩ B ( x, r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (2.9) for any n large enough depending on r and ε , we have ∀ r ∈ (cid:2) (1 − √ ε ) r, r (cid:3) ∀ δ ∈ [ δ, G n ( x, (1 − √ ε ) r, v, δ , ζ ) ⊂ G n ( x, r , v, δ , ζ + κ d M √ ε ) . Proof.
First case: x ∈ Ω . Let us assume the event G n ( x, (1 − √ ε ) r, v, δ, ζ ) occurs. Let E n be a cutsetin B ( x, (1 − √ ε ) r ) ∩ Ω n that cuts ∂ + n B ( x, (1 − √ ε ) r, v ) from ∂ − n B ( x, (1 − √ ε ) r, v ) such that E n ⊂ cyl(disc( x, (1 − √ ε ) r, v ) , δ (1 − √ ε ) r )and V ( E n ) ≤ ζα d − ((1 − √ ε ) r ) d − n d − . We define R n to be the set of edges with at least one endpoint in V (disc( x, r , v ) \ disc( x, r (1 − √ ε ) , v ) , d/n ) . By proposition 1.10, we have for n large enoughcard( R n ) ≤ d L d ( V (disc( x, r , v ) \ disc( x, r (1 − √ ε ) , v ) , d/n )) n d ≤ d H d − (cid:0) disc( x, r , v ) \ disc( x, r (1 − √ ε ) , v ) (cid:1) n d − ≤ d (cid:0) − (1 − √ ε ) d − (cid:1) α d − r d − n d − ≤ d √ εα d − r d − n d − ≤ d √ εα d − (cid:18) r − √ ε (cid:19) d − n d − ≤ d d − √ εα d − r d − n d − where we use that (1 − x ) d − ≥ − ( d − x for any x ∈ [0 ,
1] and that ε ≤ /
4. Let us prove that E n ∪ R n is a cutset in B ( x, r ) ∩ Ω n that cuts ∂ + n B ( x, r , v ) from ∂ − n B ( x, r , v ). Let γ be a path from ∂ + n B ( x, r , v )to ∂ − n B ( x, r , v ) in B ( x, r ). We write γ = ( x , e , x . . . , e p , x p ). The path γ must cross disc( x, r , v ) from22op to bottom. If γ ∩R n = ∅ then γ must cross disc( x, (1 −√ ε ) r, v ). We have γ ∩ ∂ − n B ( x, (1 −√ ε ) r, v ) = ∅ and γ ∩ ∂ + n B ( x, (1 − √ ε ) r, v ) = ∅ . We set m = inf { k ≥ x k ∈ ∂ − n B ( x, (1 − √ ε ) r, v ) } and l = sup { k ≤ m : x k ∈ ∂ + n B ( x, (1 − √ ε ) r, v ) } . The subpath of γ between l and m remains in B ( x, (1 − √ ε ) r ), it follows that γ ∩ E n = ∅ . Hence, E n ∪ R n is a cutset in B ( x, r ) ∩ Ω n that cuts ∂ + n B ( x, r , v ) from ∂ − n B ( x, r , v ). Moreover, we have E n ∪ R n ⊂ cyl(disc( x, r , v ) , δr )and V ( E n ∪ R n ) ≤ ( ζ + 10 d d − M √ ε ) α d − r d − n d − . Hence, the event G n ( x, r , v, δ, ζ + κ d M √ ε ) occurs where κ d will be chosen in such a way that κ d ≥ d d − . Second case: x ∈ ∂ ∗ Ω . We will only prove the case where x ∈ Γ since the proof for x ∈ Γ is similar.Let us assume the event G n ( x, (1 − √ ε ) r, n Ω ( x ) , δ , ζ ) occurs. Let E n be a cutset in B ( x, (1 − √ ε ) r ) ∩ Ω n that cuts ∂ + n B ( x, (1 − √ ε ) r, n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, (1 − √ ε ) r, n Ω ( x ))) from ∂ − n B ( x, (1 − √ ε ) r, n Ω ( x )) suchthat E n ⊂ cyl(disc( x, (1 − √ ε ) r, n Ω ( x )) , δ (1 − √ ε ) r )and V ( E n ) ≤ ζα d − ((1 − √ ε ) r ) d − n d − . We define R n to be the set of edges with at least one endpoint in V (( B ( x, r ) \ B ( x, r (1 − √ ε ))) ∩ ∂ ∗ Ω , d/n ) ∪ V (cid:0) ( ∂B ( x, r ) ∪ ∂B ( x, r (1 − √ ε ))) ∩ cyl(disc( x, r , n Ω ( x )) , δr ) , d/n (cid:1) , (see figure 4). Figure 4 – Representation of the sets E n and R n .23y proposition 1.10 and (2.9), we havecard( R n ) ≤ d (cid:0) H d − ( B ( x, r ) ∩ ∂ ∗ Ω) − H d − (cid:0) B ( x, r (1 − √ ε )) ∩ ∂ ∗ Ω (cid:1) + 8 δα d − r d − (cid:1) n d − ≤ d (cid:18) (1 + ε ) − (1 − ε )(1 − √ ε ) d − + 8 δ α d − α d − (cid:19) α d − r d − n d − ≤ d (cid:18) ε + ( d − √ ε + 8 ε α d − α d − (cid:19) α d − r d − n d − ≤ d d − √ ε (cid:18) d + 1 + 8 α d − α d − (cid:19) α d − r d − n d − where we recall that δ ≤ ε and ε ≤ /
4. Let us prove that E n ∪ R n is a cutset in B ( x, r ) ∩ Ω n that cuts ∂ + n B ( x, r , n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, r , n Ω ( x ))) from ∂ − n B ( x, r , n Ω ( x )) (see figure 4). Let γ be apath from ∂ + n B ( x, r , n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, r , n Ω ( x ))) to ∂ − n B ( x, r , n Ω ( x )) in B ( x, r ) ∩ Ω n . We write γ = ( x , e , x . . . , e p , x p ). . Let us assume that x ∈ Γ n . If x ∈ Γ n \ B ( x, (1 − √ ε ) r ) then x ∈ V (( B ( x, r ) \ B ( x, r (1 − √ ε ))) ∩ ∂ ∗ Ω , d/n ) and γ ∩ R n = ∅ . If x ∈ Γ n ∩ B ( x, (1 − √ ε ) r ), set m = inf { k ≥ x k ∈ ∂ n B ( x, (1 − √ ε ) r ) } . If x m ∈ ∂ + n B ( x, (1 − √ ε ) r, n Ω ( x )), then x m ∈ Ω n ∩ B + ( x, r , n Ω ( x )) ⊂ V (Ω ∩ B + ( x, r , n Ω ( x )) , d/n )and by (2.8) we have x m ∈ V ( ∂B ( x, r (1 − √ ε )) ∩ cyl(disc( x, r , n Ω ( x )) , δr ) , d/n ) . Hence, γ ∩ R n = ∅ . If x m ∈ ∂ − n B ( x, (1 − √ ε ) r, n Ω ( x )), then E n ∩ γ = ∅ by definition of the cutset E n . . Let us assume that x ∈ ∂ + n B ( x, r , n Ω ( x )) ∩ Ω n , then x ∈ Ω n ∩ B + ( x, r , n Ω ( x )) ⊂ V (Ω ∩ B + ( x, r , n Ω ( x )) , d/n )and by (2.8), we have x ∈ V ( ∂B ( x, r (1 − √ ε )) ∩ cyl(disc( x, r , n Ω ( x )) , δr ) , d/n ). It follows that γ ∩ R n = ∅ ,Hence, E n ∪ R n is a cutset in B ( x, r ) ∩ Ω n that cuts ∂ + n B ( x, r , n Ω ( x )) ∪ (Γ n \ ∂ − n B ( x, r , n Ω ( x )))from ∂ − n B ( x, r , n Ω ( x )). Moreover, we have E n ∪ R n ⊂ cyl(disc( x, r , n Ω ( x )) , δ r )and V ( E n ∪ R n ) ≤ ( ζ + κ d M √ ε ) α d − r d − n d − where κ d = 10 d d − (cid:18) d + 1 + 8 α d − α d − (cid:19) . The event G n ( x, r , n Ω ( x ) , δ , ζ + κ d M √ ε ) occurs. This section corresponds to the step 1 of the sketch of the proof in the introduction. The aim isto prove that the limiting objects must be measures supported on surfaces, in particular, that they arecontained in the set T M . We prove that if ( E, ν ) is not in T M , then it is very unlikely that there existsan almost minimal cutset in a small neighborhood of ( E, ν ). We prove this result by contradiction, ifwe have a lower bound on the probability of being in any neighborhood of (
E, ν ), then (
E, ν ) must bein T M . To prove such a result, we prove that ( E, ν ) satisfies all the required properties to be in the set T M . 24 roposition 3.1. Let ( E, ν ) ∈ B ( R d ) × M ( R d ) \ T M . Then, for any K > , there exists a neighborhood U of ( E, ν ) such that lim ε → lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≤ − K .
Proof.
Let (
E, ν ) ∈ B ( R d ) × M ( R d ). We assume that there exists K > U of ( E, ν ) that lim ε → lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ − K .
It is clear that the map ε P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) is non-decreasing. Hence, we get forany neighborhood U of ( E, ν ) ∀ ε > n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ − K . (3.1)We aim to prove that (
E, ν ) ∈ T M . We check one after the other that all the required properties aresatisfied. Step 1. We prove that L d ( E \ Ω) = 0 , ν ((Ω) c ) = 0 and ν (Ω) ≤ d M H d − (Γ ) . Let δ >
0. Let f be a continuous function compactly supported with its support included in (Ω) c . Let g bea continuous function compactly supported that takes its values in [0 ,
1] such that g = 1 on V (Ω , U = (cid:8) F ∈ B ( R d ) : d ( F, E ) ≤ δ (cid:9) × (cid:8) µ ∈ M ( R d ) : | µ ( f ) − ν ( f ) | ≤ δ, | ν ( g ) − µ ( g ) | ≤ δ (cid:9) . Fix ε >
0. Thanks to inequality (3.1), there exists an increasing sequence ( a n ) n ≥ such that P ( ∃E a n ∈ C a n ( ε ) : (R( E a n ) , µ a n ( E a n )) ∈ U ) > . For short, we will write n instead of a n . On the event {∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U } , we consider E n ∈ C n ( ε ) such that (R( E n ) , µ n ( E n )) ∈ U (if there are several possible choices, we pick one according toa deterministic rule). It follows that for n large enough, using proposition 1.10 L d ( E \ Ω) ≤ L d (R( E n ) \ Ω) + L d ( E \ R( E n )) ≤ V (cid:18) ∂ Ω , dn (cid:19) + L d (R( E n )∆ E ) ≤ δ and ν ( g ) ≤ µ n ( E n )( g ) + δ . (3.2)Note that since the support of f is included in Ω c , for n large enough, for any e ∈ Π n we have f ( c ( e )) = 0.It follows that µ n ( E n )( f ) = 0 and ν ( f ) ≤ µ n ( E n )( f ) + δ ≤ δ . Set F n = {h x, y i ∈ E dn : x ∈ Γ n , y ∈ Ω n } . It is clear that F n ∈ C n (Γ , Γ , Ω). Thanks to proposition 1.10, we have for n large enough V ( F n ) ≤ dM card(Γ n ) ≤ dM L d ( V (Γ , d/n )) n d ≤ d M H d − (Γ ) n d − . Besides, by definition of C n ( ε ) we have V ( E n ) ≤ V ( F n ) + εn d − ≤ d M H d − (Γ ) n d − + εn d − (3.3)and so µ n ( E n )( g ) ≤ V ( E n ) n d − ≤ d M H d − (Γ ) + ε . (3.4)25onsequently, we get using (3.2) and (3.4) ν (Ω) ≤ ν ( g ) ≤ d M H d − (Γ ) + δ + ε . Finally, by letting δ and ε go to 0, we obtain that L d ( E \ Ω) = 0 , ν (Ω) ≤ d M H d − (Γ )and ν ( f ) = 0. Since ν ( f ) = 0 for any continuous function having its support included in (Ω) c , weconclude that ν ((Ω) c ) = 0 . Step 2. We prove that P ( E, Ω) < ∞ . We will need the following lemma that is an adaptation oftheorem 1 by Zhang in [15]. We postpone its proof after the proof of this proposition.
Lemma 3.2.
There exists C > depending on G and Ω such that for any K > , there exists β depending on K , G and Ω such that for all n ≥ P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) ≤ C exp( − Kn d − ) . Let β be such that for all n ≥ P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) ≤ C exp( − Kn d − ) . (3.5)Let U be a neighborhood of E and U be a neighborhood of ν . We have for ε small enough usinginequality (3.3) P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U × U ) ≤ P ( ∃E n ∈ C n ( ε ) : R( E n ) ∈ U , card( E n ) ≤ βn d − )+ P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) . Combining the previous inequality, inequalities (3.1) and (3.5) and lemma 1.11, we getlim sup n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : R( E n ) ∈ U , card( E n ) ≤ βn d − ) ≥ lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U × U ) ≥ − K .
For n ≥
1, we set V n = { F ∈ B ( R d ) : d ( F, E ) ≤ /n ). We can build an increasing sequence of integers( a n ) n ≥ such that P ( ∃E a n (Γ a n ∪ Γ a n )-cutset such that R( E a n ) ∈ V n and card( E a n ) ≤ βa nd − ) > . We can choose a deterministic sequence of cutsets ( E a n ) n ≥ such that for any n ≥ d ( E, R( E a n )) ≤ /n .Besides, it is easy to check that P (R( E a n ) , Ω) ≤ card( E a n ) n d − ≤ β . Since the map F
7→ P ( F, Ω) is lower semi-continuous for the topology induced by d , we get P ( E, Ω) ≤ lim sup n →∞ P (R( E a n ) , Ω) ≤ β < ∞ . Step 3. We prove that for any x ∈ E lim sup r → ν ( B ( x,r )) α d − r d − ≤ ν G ( n • ( x )) . Here • corresponds to E orΩ depending on x . • Case x ∈ ∂ ∗ E ∩ Ω . Let δ ∈ ]0 , r > δ such that ∀ < r ≤ r L d (( E ∩ B ( x, r ))∆ B − ( x, r, n E ( x ))) ≤ δα d r d . (3.6)26p to choosing a smaller r , we can assume that B ( x, r ) ⊂ Ω (we recall that Ω is open). Let r ≤ r / ε >
0. Since ν ( B ( x, r )) < ∞ there exists a continuous function f taking its values in [0 ,
1] withsupport included in B ( x, r ) such that ν ( f ) ≥ ν ( B ( x, r )) − εα d − r d − . (3.7)Set U = (cid:8) F ∈ B ( R d ) : L d ( F ∆ E ) ≤ δα d r d (cid:9) × (cid:8) µ ∈ M ( R d ) : | µ ( f ) − ν ( f ) | ≤ εα d − r d − (cid:9) . Let ε >
0. Thanks to inequality (3.1), there exists an increasing sequence ( a n ) n ≥ such that P ( ∃E a n ∈ C a n ( ε ) : (R( E a n ) , µ a n ( E a n )) ∈ U ) > . For short, we will write n instead of a n . On the event {∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U } , we choose E n ∈ C n ( ε ) such that (R( E n ) , µ n ( E n )) ∈ U (if there are several possible choices, we pick one accordingto a deterministic rule). To shorten the notation, we write µ n for µ n ( E n ) and E n for R( E n ). Usinginequality (3.7), we have µ n ( B ( x, r )) ≥ µ n ( f ) ≥ ν ( f ) − εα d − r d − ≥ ν ( B ( x, r )) − εα d − r d − . (3.8)Set r = (1 + √ δ ) r . Note that r ≤ r ≤ r . Thanks to inequality (3.6), we have L d (( E n ∩ B ( x, r ))∆ B − ( x, r , n E ( x ))) ≤ L d (( E ∩ B ( x, r ))∆ B − ( x, r , n E ( x ))) + L d ( E ∆ E n ) ≤ δα d r d and for n large enough card (cid:0) ( E n ∩ B ( x, r ))∆ B − ( x, r , n E ( x )) ∩ Z dn (cid:1) ≤ δα d r d . (3.9)For each γ in { /n, . . . , ( b nr √ δ c − /n } , we define B ( γ ) = B ( x, r − γ, n E ( x )) . The sets ∂ + n B ( γ ) ∪ ∂ − n B ( γ ) are pairwise disjoint for different γ . Moreover, using inequality (3.9), wehave for n large enough X γ =1 /n,..., ( b nr √ δ c− /n card( E n ∩ ∂ + n B ( γ )) + card( E cn ∩ ∂ − n B ( γ )) ≤ card( E n ∆( B − ( x, r , v ) ∩ Z dn ) ≤ δα d r d n d . By a pigeon-hole principle, there exists γ in { /n, . . . , ( b nr √ δ c − /n } such that for n large enoughcard( E n ∩ ∂ + n B ( γ )) + card( E cn ∩ ∂ − n B ( γ )) ≤ δα d r d n d b nr √ δ c − ≤ δα d r d n d nr √ δ = 6 √ δα d r d − n d − . (3.10)If there are several possible choices for γ , we pick the smallest one. We denote by X + and X − thefollowing sets of edges: X + = {h y, z i ∈ E dn : y ∈ ∂ + n B ( γ ) ∩ E n } , X − = {h y, z i ∈ E dn : y ∈ ∂ − n B ( γ ) ∩ E cn } . Let us control the number of edges in X + ∪ X − using inequality (3.10):card( X + ∪ X − ) ≤ d (cid:0) card( E n ∩ ∂ + n B ( γ )) + card( E cn ∩ ∂ − n B ( γ )) (cid:1) ≤ d √ δα d r d − n d − . (3.11)27igure 5 – Representation of the cutset F n .Let G n be a minimal cutset for φ n ( ∂ − n B ( γ ) , ∂ + n B ( γ ) , B ( γ )). Set F n = ( E n \ B ( γ )) ∪ X + ∪ X − ∪ G n . We claim that F n is a (Γ n , Γ n )-cutset (see figure 5). Let γ be a path from Γ n to Γ n in Ω n . We write γ = ( x , e , x . . . , e p , x p ). . If γ ∩ (( E n \ B ( γ )) ∪ X + ∪ X − ) = ∅ then γ ∩ F n = ∅ . . Let us assume that γ ∩ (( E n \ B ( γ )) ∪ X + ∪ X − ) = ∅ . Since γ ∩E n = ∅ , then we have γ ∩ ( E n ∩ B ( γ )) = ∅ . Besides, we have γ ∩ E cn ∩ ∂ n B ( γ ) = ∅ and γ ∩ E n ∩ ∂ n B ( γ ) = ∅ . Set m = inf { k ≥ x k ∈ E cn ∩ ∂ n B ( γ ) } and l = sup { k ≤ m : x k ∈ E n ∩ ∂ n B ( γ ) } . Moreover, since γ ∩ ( X + ∪ X − ) = ∅ , we have x l ∈ ∂ − n B ( γ ) and x m ∈ ∂ + n B ( γ ). By construction, theportion of γ between x l and x m is strictly inside B ( γ ). Since G n is a cutset between ∂ − n B ( γ ) and ∂ + n B ( γ ), it follows that γ ∩ G n = ∅ . The set F n is a (Γ n , Γ n )-cutset. Since E n ∈ C n ( ε ), we have V ( F n ) ≥ V ( E n ) − ε n d − . Using inequality (3.8), it follows that V ( G n ) + M card( X + ∪ X − ) ≥ V ( E n ∩ B ( γ )) − ε n d − ≥ µ n ( B ( x, r )) n d − − ε n d − ≥ ν ( B ( x, r )) n d − − εα d − r d − n d − − ε n d − and using inequality (3.11) ν ( B ( x, r )) ≤ φ n ( ∂ − n B ( γ ) , ∂ + n B ( γ ) , B ( γ )) n d − + 12 dM √ δα d r d − + 2 εα d − r d − + ε . We can choose ε small enough depending on r and δ such that ε α d − r d − ≤ ε . It follows that ν ( B ( x, r )) α d − r d − ≤ φ n ( ∂ − n B ( γ ) , ∂ + n B ( γ ) , B ( γ )) α d − r d − n d − + 12 dM √ δ α d α d − + 3 ε ≤ φ n ( ∂ − n B ( γ ) , ∂ + n B ( γ ) , B ( γ )) α d − ( r − γ ) d − n d − (1 + √ δ ) d − + 12 dM √ δ α d α d − + 3 ε . κ d > ν ( B ( x, r )) α d − r d − ≥ ( ν G ( n E ( x )) + ε + κ d M √ ε )(1 + √ δ ) d − + 12 dM √ δ α d α d − + 3 ε . It follows that φ n ( ∂ − n B ( γ ) , ∂ + n B ( γ ) , B ( γ )) α d − ( r − γ ) d − n d − ≥ ν G ( n E ( x )) + ε + κ d M √ ε . Then using lemma 2.7 {∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U } ⊂ G n ( x, r − γ , n E ( x ) , , ν G ( n E ( x )) + ε + κ d M √ ε ) c ⊂ G n ( x, r (1 − √ ε ) , n E ( x ) , , ν G ( n E ( x )) + ε ) c where we used that r (1 − √ ε ) ≤ r (1 − √ δ ) ≤ r − γ ≤ r . Thanks to lemma 2.6, it yields thatlim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) = −∞ . This contradicts inequality (3.1), thus ν ( B ( x, r )) α d − r d − ≤ ( ν G ( n E ( x )) + ε + κ d M √ ε )(1 + √ δ ) d − + 12 dM √ δ α d α d − + 3 ε . It follows thatlim sup r → ν ( B ( x, r )) α d − r d − ≤ ( ν G ( n E ( x )) + ε + 10 d M √ ε )(1 + √ δ ) d − + 12 dM √ δ α d α d − + 3 ε and by letting δ and ε go to 0: lim sup r → ν ( B ( x, r )) α d − r d − ≤ ν G ( n E ( x )) . • Case x ∈ ∂ ∗ Ω ∩ ((Γ \ ∂ ∗ E ) ∪ (Γ ∩ ∂ ∗ E )) This case is handled in a similar way but with an extra issue:there is no r small enough such that B ( x, r ) ⊂ Ω. The set F n that we have built may not be includedin Π n . We will only treat the case x ∈ Γ \ ∂ ∗ E (the case x ∈ Γ ∩ ∂ ∗ E is very similar). To solve thisissue, we can pick δ small enough depending on ε (small enough such that it satisfies condition (2.7) inlemma 2.6 where η is replaced by ε ) and then by lemma 2.4, we pick r small enough depending on δ such that it satisfies for all r ∈ ]0 , r ] L d ((Ω ∩ B ( x, r ))∆ B − ( x, r, n Ω ( x ))) ≤ δα d r d , (3.12)(Ω ∩ B ( x, r, n Ω ( x )))∆ B + ( x, r, n Ω ( x )) ⊂ { y ∈ B ( x, r ) : | ( y − x ) · n Ω ( x ) | ≤ δ k y − x k } (3.13)and (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( ∂ ∗ Ω ∩ B ( x, r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (3.14)Moreover, up to choosing a smaller r we can assume that B ( x, r ) ∩ ( E ∪ Γ ) = ∅ (this is possible since d ( x, Γ ∪ E ) >
0. Note that for x ∈ Γ \ ∂ ∗ E the set E n ∩ B ( x, r ) looks like Ω ∩ B + ( x, r , n Ω ( x )).Whereas for y ∈ ∂ ∗ E ∩ Γ E n ∩ B ( y, r ) looks like B − ( y, r, n Ω ( x )). We have for r ≤ r , for n large enough,using proposition 1.10 L d (( E n ∩ B − ( x, r, n Ω ( x ))) ∪ ( V ∞ (Ω , /n ) ∩ B + ( x, r, n Ω ( x ))) ≤ L d (( E n ∆ E ) ∪ ( E ∩ B − ( x, r, n Ω ( x ))) + L d (Ω ∩ B + ( x, r, n Ω ( x ))) + L d ( V ∞ (Ω , /n ) \ Ω) ≤ L d ( E n ∆ E ) + L d ( E ∩ B − ( x, r, n Ω ( x )) + L d ((Ω ∩ B ( x, r ))∆ B − ( x, r, n Ω ( x ))) + 8 dn H d − ( ∂ Ω) ≤ δα d r d E ∩ B ( x, r ) = ∅ , andcard (cid:0)(cid:0) E n ∩ B − ( x, r, n Ω ( x )) ∩ Z dn (cid:1) ∪ (cid:0) Ω n ∩ B + ( x, r, n Ω ( x )) (cid:1)(cid:1) ≤ δα d r d n d . (3.15)By the same arguments (see inequality (3.10)), there exists γ in { /n, . . . , ( b nr √ δ c − /n } such thatcard( E n ∩ ∂ − n B ( γ )) + card( E cn ∩ Ω n ∩ ∂ + n B ( γ )) ≤ √ δα d r d − n d − . We denote by X + and X − the following set of edges: X + = {h y, z i ∈ Π n : y ∈ ∂ + n B ( γ ) ∩ Ω n } , X − = {h y, z i ∈ Π n : y ∈ ∂ − n B ( γ ) ∩ E n } . Let G n be a minimal cutset for φ n ( ∂ + n B ( γ ) ∪ (Γ n \ ∂ − n B ( γ )) , ∂ − n B ( γ ) , B ( γ ) ∩ Ω n ). Set F n = ( E n \ B ( γ )) ∪ X + ∪ X − ∪ G n . The set F n may not be included in Π n , but we claim that F n ∩ Π n is a (Γ n , Γ n )- cutset. Once we provethis result, the remaining of the proof is the same than in the case x ∈ ∂ ∗ E ∩ Ω using lemma 2.6.Let γ be a path from Γ n to Γ n in Ω n . . If γ ∩ (( E n \ B ( γ )) ∪ X + ∪ X − ) = ∅ then γ ∩ ( F n ∩ Π n ) = ∅ . . Let us assume that γ ∩ (( E n \ B ( γ )) ∪ X + ∪ X − ) = ∅ . Note that for n large enough, we haveΓ n ∩ B ( γ ) = ∅ . We aim to prove that γ ∩ G n = ∅ . As in the previous case, we can extract from γ anexcursion in B ( γ ) that starts at z ∈ ( ∂ n B ( γ ) ∪ Γ n ) ∩ E n and ends at w ∈ ∂ n B ( γ ) ∩ E cn ∩ Ω n . Since γ ∩ X + = ∅ then w ∈ ∂ − n B ( γ ). Similarly, if z ∈ ∂ n B ( γ ) since γ ∩ X − = ∅ we have z ∈ ∂ + n B ( γ ). Since G n is a cutset that cuts ∂ + n B ( γ ) ∪ (Γ n \ ∂ − n B ( γ )) from ∂ − n B ( γ ) in B ( γ ) ∩ Ω n , we have G n ∩ γ = ∅ .We conclude as in the previous case thatlim sup r → ν ( B ( x, r )) α d − r d − ≤ ν G ( n Ω ( x )) . Step 4. We prove that ν is supported on E . We recall that E = ( ∂ ∗ E ∩ Ω) ∪ ( ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E )) ∪ ( ∂ ∗ E ∩ Γ ) . Let us denote by λ the following measure λ = H d − | E . By Lebesgue decomposition theorem, there exists ν λ and ν s such that ν λ is absolutely continuous withrespect to λ , there exists a Borelian set A such that λ ( A c ) = ν s ( A ) = 0 and ν = ν λ + ν s . There exists f ∈ L ( λ ) such that ν λ = f λ . Moreover, since λ is a Radon measure, it is of Vitali type(see for instance theorem 4.3. in [9]). By theorem 2.3 in [9], for λ -almost every x we have f ( x ) = lim r → ν ( B ( x, r )) λ ( B ( x, r )) ≥ . Since f ∈ L ( λ ), for λ -almost every x we have f ( x ) = lim r → λ ( B ( x, r )) Z B ( x,r ) ∩ E f ( y ) d H d − ( y ) . We aim to prove that ν s = 0. The main issue to prove this is that the reduced boundary ∂ ∗ E may bevery different from the topological boundary ∂E , the set E may not be a "continuous cutset". However,up to modifying E on a set of null measure, we can assume that ∂E = ∂ ∗ E (see for instance remark15.3 in [8]). We recall that H d − ( E ) < ∞ . Let ε >
0. By proposition 2.3, there exists a finite family ofdisjoint closed balls ( B ( x i , r i , v i )) i ∈ I ∪ I ∪ I such that for i ∈ I , we have x i ∈ ∂ ∗ E ∩ Ω, for i ∈ I , wehave x i ∈ ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E ) and for i ∈ I , we have x i ∈ ∂ ∗ E ∩ Γ , and the following properties hold H d − ( E \ ∪ i ∈ I ∪ I ∪ I B ( x i , r i ))) ≤ ε , (3.16)30 i ∈ I ∪ I ∪ I ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( E ∩ B ( x i , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (3.17) ∀ i ∈ I ∪ I ∪ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( x i ,r i ) ∩ E f ( y ) d H d − ( y ) − ν ( B ( x i , r i )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ εα d − r d − i , (3.18) ∀ i ∈ I B ( x i , r i ) ⊂ Ω and L d (( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.19) ∀ i ∈ I d ( B ( x i , r i ) , Γ ∪ E ) > L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.20) ∀ i ∈ I d ( B ( x i , r i ) , Γ ) > , L d (( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di and L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di . (3.21)Set r min = min i ∈ I ∪ I ∪ I r i . Let g be a continuous function compactly supported with values in [0 ,
1] such that g = 1 on Ω. For i ∈ I ∪ I ∪ I , let g i be a continuous function compactly supported with values in [0 ,
1] such that g i = 1on B ( x i , r i (1 − √ ε/ g i = 0 on B ( x i , r i ) c . Hence, we have ν ( g i ) ≤ ν ( B ( x i , r i )) . (3.22)Set U = (cid:8) F ∈ B ( R d ) : L d ( F ∆ E ) ≤ εα d r dmin (cid:9) × (cid:26) µ ∈ M ( R d ) : | µ ( g ) − ν ( g ) | ≤ ε and ∀ i ∈ I ∪ I ∪ I | µ ( g i ) − ν ( g i ) | ≤ εα d − r d − i (cid:27) . Let ε > a n ) n ≥ such that P ( ∃E a n ∈ C a n ( ε ) : (R( E a n ) , µ a n ( E a n )) ∈ U ) > . For short, we will write n instead of a n . On the event {∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U } , we pick E n ∈ C n ( ε ) such that (R( E n ) , µ n ( E n )) ∈ U (if there are several possible choices, we pick one accordingto a deterministic rule). To shorten the notation, we write µ n for µ n ( E n ) and E n for R( E n ). We canprove thanks to inequalities (3.19) as in the proof of inequality (3.9) that for n large enough ∀ i ∈ I card (cid:0)(cid:0) ( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i ) (cid:1) ∩ Z dn (cid:1) ≤ εα d r di n d (3.23)and ∀ i ∈ I card (cid:0)(cid:0) ( E n ∩ B ( x i , r i ))∆ B − ( x i , r i , v i ) (cid:1) ∩ Z dn (cid:1) ≤ εα d r di n d . (3.24)For i ∈ I , thanks to (3.20), as in the proof of inequality (3.15), we havecard (cid:0) ( E n ∩ B − ( x i , r i , v i ) ∩ Z dn ) ∪ (Ω n ∩ B + ( x i , r i , v i )) (cid:1) ≤ εα d r di n d . (3.25)Using proposition 1.10 and (3.21), for i ∈ I , for n large enough L d (( E cn ∩ B − ( x i , r i , v i )) ∪ ( V ∞ (Ω , /n ) ∩ B + ( x i , r i , v i )) ≤ L d (( E n ∆ E ) ∪ ( E c ∩ B − ( x i , r i , v i )) + L d (Ω ∩ B + ( x i , r i , v i )) + L d ( V ∞ (Ω , /n ) \ Ω) ≤ L d ( E n ∆ E ) + L d (( E c ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) + L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) + 8 dn H d − ( ∂ Ω) ≤ εα d r d (cid:0) ( E cn ∩ B − ( x i , r i , v i ) ∩ Z dn ) ∪ (Ω n ∩ B + ( x i , r i , v i )) (cid:1) ≤ εα d r di n d . (3.26)For each i ∈ I ∪ I ∪ I and α in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } , we define B i ( α ) = B i ( x i , r i − α ) , . By the same arguments as in the previous step, using a pigeon-hole principle, there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } such that for i ∈ I card(( E n ∪ E ) ∩ ∂ + n B i ( α i ) ∩ Z dn ) + card(( E cn ∪ E c ) ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − . (3.27)For i ∈ I there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } such thatcard(Ω n ∩ ∂ + n B i ( α i )) + card( E n ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − , (3.28)and for i ∈ I , there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } such thatcard(Ω n ∩ ∂ + n B i ( α i )) + card( E cn ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − . (3.29)We denote by X + and X − the following set of edges: X + = [ i ∈ I {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ ( E ∪ E n ) } ∪ [ i ∈ I ∪ I {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ Ω n } ,X − = [ i ∈ I {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ ( E c ∪ E cn ) } ∪ [ i ∈ I {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ E n }∪ [ i ∈ I {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ E cn } . Let us control the number of edges in X + ∪ X − using inequalities (3.17), (3.27), (3.28) and (3.29)card( X + ∪ X − ) ≤ d α d α d − X i ∈ I ∪ I ∪ I √ εα d − r d − i n d − ≤ d √ ε α d α d − X i ∈ I ∪ I ∪ I − ε H d − ( E ∩ B ( x i , r i )) n d − ≤ d √ ε α d α d − H d − ( E ) n d − (3.30)where we use that ε ≤ /
2. Let us denote by F the set of edges with at least one endpoint in V ( E \ ∪ i ∈ I ∪ I ∪ I B i ( α i ) , d/n ). Using inequality (3.17), we have X i ∈ I ∪ I ∪ I H d − ( B ( x i , r i ) ∩ E ) − H d − ( B i ( α i ) ∩ E ) ≤ X i ∈ I ∪ I ∪ I (1 + ε ) α d − r d − i − (1 − ε ) α d − ( r i − α i ) d − ≤ X i ∈ I ∪ I ∪ I (cid:0) ε − (1 − ε )(1 − √ ε ) d − (cid:1) α d − r d − i ≤ (1 + ε − (1 − ε )(1 − ( d − √ ε )) 11 − ε H d − ( E ) ≤ d √ ε H d − ( E ) (3.31)for small enough ε depending on d . Hence, we have using proposition 1.10, inequalities (3.16) and (3.31),for n large enough,card( F ) ≤ d L d ( V ( E \ ∪ i ∈ I B i ( α i ) , d/n )) n d ≤ d (2 d √ ε H d − ( E ) + ε ) n d − . (3.32)We claim that F n = X + ∪ X − ∪ F ∪ i ∈ I ∪ I ∪ I ( E n ∩ B i ( α i )) is a (Γ n , Γ n )-cutset. Let γ be a path fromΓ n to Γ n in Ω n . We write γ = ( x , e , x . . . , e p , x p ). By definition of E n , we have x ∈ E n and x p ∈ E cn . . Let us assume that x ∈ B i ( α i ) for i ∈ I . Since for n large enough B i ( α i ) ∩ Γ n = ∅ , the path γ eventually exits the ball B i ( α i ). Let x r denotes the first point that reaches ∂ n B i ( α i ), i.e. , r = inf { k ≥ x k ∈ ∂ n B i ( α i ) } . We have x r ∈ Ω n . 32 If x r ∈ ∂ + n B ( α i ), then x r ∈ ∂ + n B ( α i ) ∩ Ω n and h x r , x r +1 i ∈ X + . · If x r ∈ ∂ − n B ( α i ) ∩ E n , then h x r , x r +1 i ∈ X − . · If x r ∈ ∂ − n B ( α i ) ∩ E cn , since x ∈ E n , then γ ∩ ( ∂ e E n ∩ B i ( α i )) = ∅ and so γ ∩ ( E n ∩ B i ( α i )) = ∅ . . Let us assume that x p ∈ B i ( α i ) for i ∈ I . For n large enough, we have Γ n ∩ B i ( α i ) = ∅ , and so x / ∈ B i ( α i ). Let x l denotes the last point to enter in B i ( α i ), i.e. , l = sup { k ≥ x k ∈ ∂ n B i ( α i ) } . We have x l ∈ Ω n . · If x l ∈ ∂ + n B ( α i ), then x l ∈ ∂ + n B ( α i ) ∩ Ω n and h x l − , x l i ∈ X + . · If x l ∈ ∂ − n B ( α i ) ∩ E cn , then h x l − , x l i ∈ X − . · If x l ∈ ∂ − n B ( α i ) ∩ E n , since x p ∈ E cn , then γ ∩ ( ∂ e E n ∩ B i ( α i )) = ∅ and so γ ∩ ( E n ∩ B i ( α i )) = ∅ . . Let us assume x ∈ V (Γ \ ∂ ∗ E, d/n ) \ S i ∈ I B i ( α i ) then h x , x i ∩ F = ∅ . . Let us assume x p ∈ V (Γ ∩ ∂ ∗ E, d/n ) \ S i ∈ I B i ( α i ) then h x p − , x p i ∩ F = ∅ . . Let us assume x ∈ V (Γ \ ∂ ∗ E, d/n ) c and x p ∈ V (Γ ∩ ∂ ∗ E, d/n ) c . Set e E = E ∪ ( V ( E, d/n ) \ Ω). Itfollows that x ∈ e E and x p ∈ e E c . Let us assume there exists an excursion of γ inside a ball B i ( α i ), for i ∈ I that starts at x r ∈ ∂ n B i ( α i ) ∩ e E = ∂ n B i ( α i ) ∩ E and exits at x m ∈ ∂ n B i ( α i ) ∩ e E c = ∂ n B i ( α i ) ∩ E c (where we use that B i ( α i ) ⊂ Ω). · If x r / ∈ ∂ − n B i ( α i ) ∩ E n , then x r ∈ ∂ + n B i ( α i ) ∩ E or x r ∈ ∂ − n B i ( α i ) ∩ E cn . It follows that h x r − , x r i ∈ X − ∪ X + . · If x m / ∈ ∂ + n B i ( α i ) ∩ E cn , then x r ∈ ∂ − n B i ( α i ) ∩ E c or x r ∈ ∂ + n B i ( α i ) ∩ E n . It follows that h x m , x m +1 i ∈ X − ∪ X + . · If x r ∈ E n and x m ∈ E cn , then γ ∩ ( E n ∩ B i ( α i )) = ∅ .If no such excursion exists then the path γ must cross ( ∂ e E \ ∂ ( V (Ω , d/n ))) \ ∪ i ∈ I B i ( α i ). We distinguishtwo cases. · If γ crosses ( ∂E ∩ Ω) \ ∪ i ∈ I B i ( α i ). Since ∂E = ∂ ∗ E , it follows that γ ∩ F = ∅ . · If γ crosses ∂ e E outside Ω. Then γ must cross V ((Γ \ ∂ ∗ E ) \ ∪ i ∈ I B i ( α i ) , d/n ) or V ((Γ ∩ ∂ ∗ E ) \∪ i ∈ I B i ( α i ) , d/n ). Hence, γ ∩ F = ∅ .For all these cases, we have γ ∩ F n = ∅ , thus F n is indeed a (Γ n , Γ n )-cutset. Since E n ∈ C n ( ε ), usinginequalities (3.30) and (3.32), we have V ( E n ) ≤ V ( F n ) + ε n d − ≤ V ( X + ∪ X − ∪ F ∪ i ∈ I ∪ I ∪ I ( E n ∩ B i ( α i ))) + ε n d − ≤ M (card( X + ∪ X − ) + card( F )) + X i ∈ I ∪ I ∪ I V ( E n ∩ B i ( α i )) + ε n d − ≤ M (cid:18)(cid:18) d α d α d − + 20 d (cid:19) √ ε H d − ( E ) + 10 d ε (cid:19) n d − + n d − X i ∈ I ∪ I ∪ I µ n ( g i ) + ε n d − (3.33)where we recall that by construction of g i and α i , we have g i = 1 on B i ( α i ). Using the fact that( E n , µ n ) ∈ U , inequalities (3.17), (3.18), and (3.22), it follows that X i ∈ I ∪ I ∪ I µ n ( g i ) ≤ X i ∈ I ∪ I ∪ I (cid:0) ν ( g i ) + εα d − r d − i (cid:1) ≤ X i ∈ I ∪ I ∪ I (cid:0) ν ( B ( x i , r i )) + εα d − r d − i (cid:1) ≤ X i ∈ I ∪ I ∪ I Z B ( x i ,r i ) ∩ E f ( y ) d H d − ( y ) + 2 εα d − r d − i ! ≤ Z E f ( y ) d H d − ( y ) + 2 ε − ε H d − ( E ) . ν ( R d ) ≤ ν ( g ) ≤ µ n ( g ) + ε = V ( E n ) /n d − + ε . Combining all the three previous inequalities, we get ν ( R d ) ≤ M (cid:18)(cid:18) d α d α d − + 20 d (cid:19) √ ε H d − ( E ) + 10 d ε (cid:19) + Z E f ( y ) d H d − ( y ) + 2 ε − ε H d − ( E ) + ε + ε . By letting first ε go to 0 and then ε go to 0, we obtain that ν ( R d ) ≤ Z E f ( y ) d H d − ( y ) . Since ν ( R d ) ≥ ν λ ( R d ) = R E f ( y ) d H d − ( y ), we obtain that ν ( R d ) = Z E f ( y ) d H d − ( y ) . Thus, we have ν s = 0 and ν is absolutely continuous with respect to λ . Step 5. We prove that ( E, ν ) is minimal. Let F ∈ B ( R d ) such that F ⊂ Ω and P ( F, Ω) < ∞ . Set F = ( ∂ ∗ F ∩ Ω) ∪ ( ∂ ∗ Ω ∩ (Γ \ ∂ ∗ F ) ∪ ( ∂ ∗ F ∩ Γ ) . Let us denote by F and F the following sets: F = F \ E and F = F ∩ E . Note that if x ∈ F , then d ( x, F ) >
0. Indeed, if d ( x, F ) = 0 then x ∈ E and this contradicts the factthat x ∈ F . Let ε >
0. Let δ be small enough depending on ε as prescribed in inequality (2.7) in lemma2.6. By proposition 2.3, there exists a finite family of disjoint closed balls ( B ( x i , r i , v i )) i ∈ I ∪ I ∪ I of radiussmaller than ε such that for i ∈ I , we have x i ∈ ∂ ∗ F ∩ Ω, for i ∈ I , we have x i ∈ ∂ ∗ Ω ∩ (Γ \ ∂ ∗ F )and for i ∈ I , we have x i ∈ ∂ ∗ F ∩ Γ . Moreover, we have that for any i ∈ I ∪ I ∪ I , if x i ∈ F , then B ( x i , r i ) ∩ F = ∅ , and the following properties hold: H d − ( F \ ∪ i ∈ I ∪ I ∪ I B ( x i , r i ))) ≤ ε , (3.34) ∀ i ∈ I ∪ I ∪ I ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( F ∩ B ( x i , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (3.35) ∀ i ∈ I B ( x i , r i ) ⊂ Ω and L d (( F ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.36) ∀ i ∈ I such that x i ∈ F B ( x i , r i ) ⊂ Ω and L d (( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.37) ∀ i ∈ I d ( B ( x i , r i ) , F ∪ Γ ) > L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.38) ∀ i ∈ I d ( B ( x i , r i ) , Γ ) > , L d (( F ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di and L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (3.39) ∀ i ∈ I ∪ I (Ω ∩ B ( x i , r i , v i ))∆ B − ( x i , r i , v i )) ⊂ { y ∈ B ( x i , r i ) : | ( y − x i ) · v i | ≤ δ k y − x i k } , (3.40) ∀ i ∈ I ∪ I ∪ I ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z F ∩ B ( x i ,r ) ν G ( n • ( y )) d H d − ( y ) − ν G ( v i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (3.41)34here • = F if x i ∈ ∂ ∗ F and • = Ω if x i ∈ ∂ ∗ Ω \ ∂ ∗ F . Set r min = min i ∈ I ∪ I ∪ I r i . Let g be a continuous function compactly supported such that g = 1 on Ω. For i ∈ I ∪ I ∪ I , let g i bea continuous function compactly supported with values in [0 ,
1] such that g i = 1 on B ( x i , r i (1 − √ ε/ g i = 0 on B ( x i , r i ) c . Hence, we have ν ( g i ) ≤ ν ( B ( x i , r i )) . (3.42)Set U = (cid:8) F ∈ B ( R d ) : L d ( F ∆ E ) ≤ εα d r dmin (cid:9) × µ ∈ M ( R d ) : ∀ i ∈ I ∪ I ∪ I | µ ( g i ) − ν ( g i ) | ≤ εα d − r d − i , | µ ( g ) − ν ( g ) | ≤ ε . Let ε > n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ − K .
For i ∈ I ∪ I and x i ∈ F , by the same arguments as in the previous step, there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } such thatcard( F ∩ ∂ + n B i ( α i ) ∩ Z dn ) + card( F c ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − . (3.43)For i ∈ I such that x i ∈ F , there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c − /n } such thatcard(Ω n ∩ ∂ + n B i ( α i )) + card(Ω cn ∩ ∂ − n B i ( α i )) ≤ √ εα d r d − i n d − . (3.44)The values of α i are deterministic and depend on n and ε . Let us denote by E the following event E = \ i ∈ I ∪ I ∪ I : x i ∈ F G n (cid:0) x i , r i − α i , v i , , ν G ( v i ) + ε + κ d M √ ε (cid:1) where κ d is given by lemma 2.7. By lemma 2.7, we have P ( E c ) ≤ X i ∈ I ∪ I ∪ I : x i ∈ F P (cid:0) G n (cid:0) x i , r i − α i , v i , , ν G ( v i ) + ε + κ d M √ ε (cid:1) c (cid:1) ≤ X i ∈ I ∪ I ∪ I : x i ∈ F P (cid:0) G n (cid:0) x i , r i (1 − √ ε ) , v i , , ν G ( v i ) + ε (cid:1) c (cid:1) . By lemma 2.6, we have lim sup n →∞ n d − log P ( E c ) = −∞ . Besides, we havelim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U, E c ) ≤ lim sup n →∞ n d − log P ( E c ) = −∞ . Hence, by lemma 1.11,lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U, E ) ≥ − K .
Thus, there exists an increasing sequence ( a n ) n ≥ such that P ( ∃E a n ∈ C a n ( ε ) : (R( E a n ) , µ a n ( E a n )) ∈ U, E ) > . n instead of a n . On the event {∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U, E} , we choose E n ∈ C n ( ε ) such that (R( E n ) , µ n ( E n )) ∈ U (if there are several possible choices, we pick one accordingto a deterministic rule). To shorten the notation, we write µ n for µ n ( E n ) and E n for R( E n ). As in theprevious step, for i ∈ I ∪ I ∪ I such that x i ∈ F , there exists α i in {b nr i √ ε/ c /n, . . . , ( b nr i √ ε c− /n } such that if i ∈ I thencard(( E n ∪ F ) ∩ ∂ + n B i ( α i ) ∩ Z dn ) + card(( E cn ∪ F c ) ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − , (3.45)if i ∈ I , then card(Ω n ∩ ∂ + n B i ( α i )) + card( E n ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − , (3.46)finally, if i ∈ I , thencard(Ω n ∩ ∂ + n B i ( α i ) ∩ Z dn ) + card( E cn ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d r d − i n d − . (3.47)We denote by X +1 , X +2 , X − and X − the following set of edges: X +1 = [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ F } ∪ [ i ∈ I ∪ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ Ω n } ,X +2 = [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ ( F ∪ E n ) } ∪ [ i ∈ I ∪ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ Ω n } ,X − = [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ F c } ∪ [ i ∈ I ∪ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ Ω c } ,X − = [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ ( F c ∪ E cn ) } ∪ [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ E n }∪ [ i ∈ I : x i ∈ F {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ E cn } . We can control the number of edges in X +1 ∪ X +2 ∪ X − ∪ X − by the same arguments as in the previousstep card( X +1 ∪ X +2 ∪ X − ∪ X − ) ≤ d α d α d − √ ε H d − ( F ) . (3.48)Let us denote by F the set of edges with at least one endpoint in V ( F \ ∪ i ∈ I B i ( α i ) , d/n ). We have bythe same computations as in (3.32)card( F ) ≤ d (2 d √ ε H d − ( F ) + ε ) n d − ≤ d √ ε H d − ( F ) n d − (3.49)for ε small enough depending on d and F . On the event E , for i ∈ I ∪ I ∪ I such that x i ∈ F , wedenote by G in the cutset in the definition of the event G n ( x i , r i − α i , v i , , ν G ( v i ) + ε + κ d M √ ε ) (if thereare several possible choices, we choose one according to a deterministic rule). We claim that F n = X +1 ∪ X +2 ∪ X − ∪ X − ∪ F [ i ∈ I ∪ I ∪ I : x i ∈ F ( E n ∩ B i ( α i )) [ i ∈ I ∪ I ∪ I : x i ∈ F G in is a (Γ n , Γ n )-cutset.Let γ be a path from Γ n to Γ n in Ω n . We write γ = ( x , e , x . . . , e p , x p ). . Let us assume that x ∈ B i ( α i ) for i ∈ I and x i ∈ F or x p ∈ B j ( α j ) for j ∈ I and x j ∈ F . Then,by the same arguments as in the previous step, we have F n ∩ γ = ∅ . . Let us assume that x ∈ B i ( α i ) for i ∈ I and x i ∈ F . If x ∈ ∂ − n B i ( α i ), then x ∈ ∂ − n B i ( α i ) ∩ Ω c and h x , x i ∈ X − . Let us assume x / ∈ ∂ − n B i ( α i ). Since for n large enough B i ( α i ) ∩ Γ n = ∅ , the path γ eventually exits the ball B i ( α i ). Set r = inf { k ≥ x k ∈ ∂ n B i ( α i ) } . We have x r ∈ Ω n . 36 If x r ∈ ∂ + n B i ( α i ), then x r ∈ ∂ + n B i ( α i ) ∩ Ω n and h x r , x r +1 i ∈ X +1 . · If x r ∈ ∂ − n B i ( α i ), then since G in is a cutset between Γ n \ ∂ − n B i ( α i ) and ∂ − n B i ( α i ) in B i ( α i ) then γ ∩ G in = ∅ . . Let us assume that x p ∈ B i ( α i ) for i ∈ I and x i ∈ F . By the same reasoning as in the previous case,we can prove that γ ∩ F n = ∅ . . Let us assume x ∈ V ∞ (Γ \ ∂ ∗ F , /n ) \ S i ∈ I B i ( α i ) then h x , x i ∩ F = ∅ . . Let us assume x p ∈ V ∞ (Γ ∩ ∂ ∗ E, /n ) \ S i ∈ I B i ( α i ) then h x p − , x p i ∩ F = ∅ . . Let us assume x ∈ V ∞ (Γ \ ∂ ∗ F , /n ) c and x p ∈ V ∞ (Γ ∩ ∂ ∗ F, /n ) c . Set e F = F ∪ ( V ∞ ( F, /n ) \ Ω).It follows that x ∈ e F and x p ∈ e F c . Let us assume there exists an excursion of γ inside a ball B i ( α i ), for i ∈ I and x i ∈ F that starts at x r ∈ ∂ n B i ( α i ) ∩ e F = ∂ n B i ( α i ) ∩ F and exits at x m ∈ ∂ n B i ( α i ) ∩ e F c = ∂ n B i ( α i ) ∩ F c , where we use that B i ( α i ) ⊂ Ω. · If x r ∈ ∂ + n B i ( α i ) ∩ F , then h x r − , x r i ∈ X +1 . · If x m ∈ ∂ − n B i ( α i ) ∩ F c , then h x m , x m +1 i ∈ X − . · If x r ∈ ∂ − n B i ( α i ) and x m ∈ ∂ + n B i ( α i ), then since G in cuts ∂ − n B i ( α i ) from ∂ + n B i ( α i ) in B i ( α i ), wehave γ ∩ G in = ∅ .Let us assume there exists an excursion of γ inside a ball B i ( α i ), for i ∈ I and x i ∈ F that starts at x r ∈ ∂ n B i ( α i ) ∩ F and exits at x m ∈ ∂ n B i ( α i ) ∩ F c . · If x r ∈ ∂ + n B i ( α i ) ∩ F , then h x r − , x r i ∈ X +2 . If x r ∈ ∂ − n B i ( α i ) ∩ E cn , then h x r − , x r i ∈ X − . · If x m ∈ ∂ − n B i ( α i ) ∩ F c , then h x m , x m +1 i ∈ X − . If x m ∈ ∂ + n B i ( α i ) ∩ E n , then h x m − , x m i ∈ X +2 . · Finally, if x r ∈ E n and x m ∈ E cn , then γ ∩ ( E n ∩ B i ( α i )) = ∅ .If no such excursion exists then the path γ must cross ( ∂ e F \ ∂ V ∞ (Ω , \∪ i ∈ I B i ( α i ). We conclude asin the previous step that γ ∩ F = ∅ . For all these cases, we have γ ∩ F n = ∅ , thus F n is a (Γ n , Γ n )-cutset.Note that B i ( α i ) ⊂ B ( x i , r i (1 − √ ε/ g i = 1 on B i ( α i ). On the event E , using inequalities(3.35) and (3.41), we have1 n d − V [ i ∈ I ∪ I ∪ I : x i ∈ F ( E n ∩ B i ( α i )) [ i ∈ I ∪ I ∪ I : x i ∈ F G in ≤ n d − X i ∈ I ∪ I ∪ I : x i ∈ F V ( E n ∩ B i ( α i )) + 1 n d − X i ∈ I ∪ I ∪ I : x i ∈ F V ( G in ) ≤ X i ∈ I ∪ I ∪ I : x i ∈ F µ n ( g i ) + X i ∈ I ∪ I ∪ I : x i ∈ F (cid:0) ν G ( v i ) + ε + M κ d √ ε (cid:1) α d − r d − i ≤ X i ∈ I ∪ I ∪ I : x i ∈ F (cid:0) ν ( B ( x i , r i )) + εα d − r d − i (cid:1) + X i ∈ I ∪ I ∪ I : x i ∈ F Z F ∩ B ( x i ,r i ) ν G ( n • ( y )) d H d − ( y ) + 2 M κ d √ εα d − r d − i ! ≤ Z V ( F ,ε ) ∩ E f ( y ) d H d − ( y ) + Z F ν G ( n • ( y )) d H d − ( y ) + 4 M κ d √ ε H d − ( F ) . (3.50)for small enough ε depending on d and M . We have used the two following facts:(i) for i ∈ I ∪ I ∪ I such that x i ∈ F , we have by construction of the covering F ∩ B ( x i , r i ) = F ∩ B ( x i , r i ) , (ii) for x i ∈ F , since r i ≤ ε , we have F ∩ B ( x i , r i ) ⊂ V ( F , ε ) ∩ B ( x i , r i ) . V ( E n ) = n d − µ n ( g ) ≥ n d − ( ν ( g ) − ε ) = n d − (cid:18)Z E f ( y ) d H d − ( y ) − ε (cid:19) . (3.51)Using that E n ∈ C n ( ε ), inequalities (3.48), (3.49), (3.50) and (3.51), it follows that Z E f ( y ) d H d − ( y ) − ε ≤ κ √ ε H d − ( F ) + Z V ( F ,ε ) ∩ E f ( y ) d H d − ( y ) + Z F ν G ( n • ( y )) d H d − ( y ) + ε where κ depends on d and M . Note that by the dominated convergence theorem, we havelim ε → Z V ( F ,ε ) ∩ E f ( y ) d H d − ( y ) = Z F ∩ E f ( y ) d H d − ( y ) = Z F ∩ E f ( y ) d H d − ( y ) . By letting first ε go to 0, then δ go to 0 and finally ε go to 0, we obtain that Z E f ( y ) d H d − ( y ) ≤ Z F ∩ E f ( y ) d H d − ( y ) + Z F \ E ν G ( n • ( y )) d H d − ( y ) . The minimality of (
E, f ) follows.
Proof of lemma 3.2.
This proof is just an adaptation of the proof of Zhang in [15]. In [15], Zhang controlsthe cardinal of a minimal cutset that cuts a given box from infinity. Actually, his proof may be adaptedto any connected cutset (not necessarily minimal) with a control on its capacity. In our context, wehave the following control on the capacity V ( E n ) ≤ d M H d − (Γ ) n d − . We need the cutset to beconnected to be able to upper-bound the number of possible realizations of the cutset E n . To solve thisissue, we do as in remark 19 in [4], we consider the union of E n with the edges that lie along Γ: it isalways connected, and the number of edges we have added is upper-bounded by cn d − for a constant c that depends only on the domain Ω since Γ is piecewise of class C (see proposition 1.10). Let K ≥ C and C such that ∀ n ≥ P ( ∃E n (Γ n ∪ Γ n )-cutset such that V ( E n ) ≤ d M H d − (Γ ) n d − and card( E n ) ≥ βn d − ) ≤ C exp( − C βn d − ) . We set β = K/C . The result follows. This section corresponds to the step 2 of the sketch of the proof in the introduction. For (
E, ν ) ∈ T M such that e I ( E, ν ) < ∞ , we cover almost E by a family of disjoint balls. In each ball, we build a cutsetthat is almost flat and has almost the same local capacity than ( E, ν ), the cost of this operation maybe controlled by lower large deviations for the maximal flow in balls (see lemma 2.6). We then fill theholes by adding a negligible amount of edges to merge all these cutsets inside the balls into a cutset in C n (Γ , Γ , Ω). The main technical difficulty is to ensure that the cutset we have built is almost minimal.To do so, we have to ensure that anywhere outside a small region around E , the local maximal flow isnot abnormally low. Proposition 4.1 (Lower bound) . Let ( E, ν ) ∈ T M such that e I ( E, ν ) < ∞ . Then, for any neighborhood U of ( E, ν ) , we have lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ − e I ( E, ν ) . Proof.
Let (
E, ν ) ∈ T M such that e I ( E, ν ) < ∞ . We write ν = f H d − | E . Let U be a neighborhoodof ( E, ν ) for the product topology O × O . There exists E , ξ >
0, there exist p continuous functions g , . . . , g p having compact support and ξ > U = { F ∈ B ( R d ) : L d ( F ∆ E ) ≤ ξ } × (cid:8) ρ ∈ M ( R d ) : ∀ i ∈ { , . . . , p } | ρ ( g i ) − ν ( g i ) | ≤ ξ (cid:9) ⊂ U . g , . . . , g p are uniformly continuous. Let η >
0. Let ε >
0, we will choose later dependingon η . There exists δ > ∀ i ∈ { , . . . , p } ∀ x, y ∈ R d k x − y k ≤ δ = ⇒ | g i ( x ) − g i ( y ) | ≤ ε . (4.1)Thanks to lemma 3.2, there exists β > n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) ≤ − e I ( E, ν ) . We define the event E (0) = (cid:8) ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − (cid:9) c . Step 1. Build a covering of C β . Let F ∈ C β where we recall that C β was defined in (1.10). Bydominated convergence theorem, we havelim δ → Z ( F \V ( E ,δ )) ν G ( n • ( y )) d H d − ( y ) = Z ( F \ E ) ν G ( n • ( y )) d H d − ( y ) . Consequently, there exists δ F > Z ( F \V ( E ,δ F )) ν G ( n • ( y )) d H d − ( y ) ≥ Z ( F \ E ) ν G ( n • ( y )) d H d − ( y ) − ε . (4.2)We recall that the notation n • ( x ) is to lighten the expressions, • will correspond to E , F or Ω dependingon x . By proposition 2.3, there exists a finite covering ( B ( x Fi , r Fi , v Fi )) i ∈ I F of ( F \ V ( E , δ F )) ∪ ( E ∩ F )such that r Fi ≤ δ F / i ∈ I F , H d − ((( F \ V ( E , δ F )) ∪ ( E ∩ F )) \ ∪ i ∈ I F B ( x Fi , r Fi )) ≤ ε , (4.3) ∀ i ∈ I F ∀ < r ≤ r Fi (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( F ∩ B ( x Fi , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε and if x Fi ∈ F ∩ E ∀ < r ≤ r Fi (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( E ∩ B ( x Fi , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (4.4) ∀ i ∈ I F such that x Fi ∈ ∂ ∗ F ∩ Ω B ( x Fi , r Fi ) ⊂ Ω and L d (( F ∩ B ( x Fi , r Fi ))∆ B − ( x Fi , r Fi , v Fi )) ≤ εα d ( r Fi ) d , (4.5) ∀ i ∈ I F such that x Fi ∈ Γ \ ∂ ∗ F d ( B ( x Fi , r Fi ) , F ∪ Γ ) > L d ((Ω ∩ B ( x Fi , r Fi ))∆ B − ( x Fi , r Fi , v Fi )) ≤ εα d ( r Fi ) d , (4.6) ∀ i ∈ I F such that x Fi ∈ ∂ ∗ Ω ∩ Γ d ( B ( x Fi , r Fi ) , Γ ) > L d (( F ∩ B ( x Fi , r Fi ))∆ B − ( x Fi , r Fi , v Fi )) ≤ εα d ( r Fi ) d , (4.7) ∀ i ∈ I F such that x Fi ∈ F \ E ∀ < r ≤ r Fi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z F ∩ B ( x Fi ,r ) ν G ( n • ( y )) d H d − ( y ) − ν G ( v Fi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (4.8) ∀ i ∈ I F such that x Fi ∈ F ∩ E ∀ < r ≤ r Fi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z E ∩ B ( x Fi ,r ) f ( y ) d H d − ( y ) − f ( x Fi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (4.9)39 i ∈ I F such that x Fi ∈ F ∩ E ∀ < r ≤ r Fi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z E ∩ B ( x Fi ,r ) J n • ( y ) ( f ( y )) d H d − ( y ) − J v Fi ( f ( x Fi )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (4.10)Set ε F = εα d min i ∈ I F ( r Fi ) d . Since C β is compact, we can extract from ( B d ( F, ε F ) , F ∈ C β ) a finite covering ( B d ( F, ε F ) , F ∈ A ) of C β with card( A ) < ∞ . Since e I ( E, ν ) < ∞ and H d − ( E ) < ∞ , we can choose δ small enough such that δ ≤
14 min F ∈A δ F , ∀ F ∈ A ∀ j ∈ I F such that x Fj ∈ F ∩ E Z E ∩ ( B ( x Fj ,r Fj + δ ) \ B ( x Fj ,r Fj )) J n • ( x ) ( f ( x )) d H d − ( x ) ≤ εα d − ( r Fj ) d − and H d − (cid:0) E ∩ (cid:0) B ( x Fj , r Fj + δ ) \ B ( x Fj , r Fj ) (cid:1)(cid:1) ≤ εα d − ( r Fj ) d − . (4.11)Set r min = min F ∈A min i ∈ I F r Fi . Step 2. We build a covering of E that depends on the covering of C β . By proposition 2.3, thereexists a finite covering ( B ( x Ei , r Ei , v Ei )) i ∈ I E of E such that for all i ∈ I E , r Ei ≤ min( r min , δ , δ ) and thefollowing properties hold: H d − E \ [ i ∈ I E B ( x Ei , r Ei ) ! ≤ ε , (4.12) ∀ i ∈ I E ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( E ∩ B ( x Ei , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (4.13) ∀ i ∈ I E such that x Ei ∈ ∂ ∗ E ∩ Ω B ( x Ei , r Ei ) ⊂ Ω and L d (( E ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ≤ εα d ( r Ei ) d , (4.14) ∀ i ∈ I E such that x Ei ∈ Γ \ ∂ ∗ E d ( B ( x Ei , r Ei ) , Γ ∪ E ) > L d ((Ω ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ≤ εα d ( r Ei ) d , (4.15) ∀ i ∈ I E such that x Ei ∈ ∂ ∗ E ∩ Γ d ( B ( x Ei , r Ei ) , Γ ) > , L d (( E ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ≤ εα d ( r Ei ) d (4.16)and L d ((Ω ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ≤ εα d ( r Ei ) d , (4.17) ∀ i ∈ I E such that x Ei ∈ ∂ ∗ Ω (Ω ∩ B ( x Ei , r Ei )∆ B + ( x Ei , r Ei , v Ei )) ⊂ (cid:8) y ∈ B ( x Ei , r Ei ) : | ( y − x Ei ) · v Ei | ≤ δ k y − x Ei k (cid:9) , (4.18) ∀ i ∈ I E ∀ < r ≤ r Ei (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z E ∩ B ( x Ei ,r ) f ( y ) d H d − ( y ) − f ( x Ei ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (4.19) ∀ i ∈ I E ∀ < r ≤ r Ei (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α d − r d − Z E ∩ B ( x Ei ,r ) J n • ( y ) ( f ( y )) d H d − ( y ) − J v Ei ( f ( x Ei )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (4.20)40e need to slightly reduce the r Ei . Using inequality (4.14), for i ∈ I E and x Ei ∈ E \ ∂ ∗ Ω, by the samearguments as in the previous section, there exists α i in { /n, . . . , ( b nr Ei √ ε c − /n } such thatcard( E ∩ ∂ + n B i ( α i ) ∩ Z dn ) + card( E c ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d ( r Ei ) d − n d − , (4.21)where B i ( α i ) = B ( x Ei , r Ei − α i , v Ei ). Using inequalities (4.15) and (4.16), for i ∈ I E such that x i ∈ ∂ ∗ Ω,there exists α i in { /n, . . . , ( b nr Ei √ ε c − /n } such thatcard(Ω n ∩ ∂ + n B i ( α i )) + card(Ω c ∩ ∂ − n B i ( α i ) ∩ Z dn ) ≤ √ εα d ( r Ei ) d − n d − . (4.22)The values of α i are deterministic and depend on n . We denote by X + and X − the following set ofedges: X + = [ i ∈ I E : x Ei ∈ E \ ∂ ∗ Ω {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ E } ∪ [ i ∈ I E : x Ei ∈ ∂ ∗ Ω {h y, z i ∈ E dn : y ∈ ∂ + n B i ( α i ) ∩ Ω n } ,X − = [ i ∈ I E : x Ei ∈ E \ ∂ ∗ Ω {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ E c } ∪ [ i ∈ I E : x Ei ∈ ∂ ∗ Ω {h y, z i ∈ E dn : y ∈ ∂ − n B i ( α i ) ∩ Ω c } . By similar computations as in (3.30), using (4.13), we havecard( X − ∪ X + ) ≤ d α d α d − √ ε H d − ( E ) n d − ≤ ηn d − (4.23)for ε small enough depending on η , d and E . Step 3. We build a configuration on which the event {∃E n ∈ C n ( u ( η )) : (R( E n ) , µ n ( E n )) ∈ U } occurs. Set ε = α d α d − − d − ε and ¯ r Ei = (1 − √ ε ) r Ei , for i ∈ I E . Set E (1) = \ F ∈A \ i ∈ I F : x Fi ∈ F \ E G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi )) c , E (2) = \ F ∈A \ i ∈ I F : x Fi ∈ F ∩ E , f ( x Fi ) ≥ η G n ( x Fi , r Fi , v Fi , ε, (1 − η ) f ( x Fi )) c , E (3) = \ i ∈ I E : f ( x Ei ) ≥ η G n ( x Ei , ¯ r Ei , v Ei , ε, (1 − η ) f ( x Ei )) c and E (4) = \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) . The first event ensures that the flow is not abnormally low outside E . The three other events ensurethat the local flow near E is close to f . For any F ∈ A , for any i ∈ I F such that x Fi ∈ F \ E , we have byconstruction x Fi ∈ F \ V ( E , δ F ), r Fi ≤ δ F / j ∈ I E , r Ej ≤ δ F /
4. It follows that B ( x Fi , r Fi ) ∩ [ j ∈ I E B ( x Ej , ¯ r Ej ) = ∅ . Hence, the event E (1) is independent of the event E (4) . We claim that on the event E (0) ∩ E (1) ∩ E (2) ∩E (3) ∩ E (4) , the following event occurs {∃E n ∈ C n ( u ( η )) : (R( E n ) , µ n ( E n )) ∈ U } u that goes to 0 when η goes to 0. We build the cutset.
By lemma 2.7, we have G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ⊂ G n ( x Ei , r Ei − α i , v Ei , ε , f ( x Ei ) + η + M κ d √ ε ) where κ d ≥ E (4) , for any i ∈ I E ,denote by E n ( i ) the cutset corresponding to the event G n ( x Ei , r Ei − α i , v Ei , ε , f ( x Ei )+ η + M κ d √ ε ). Denoteby F the edges that have at least one endpoint in V E \ [ i ∈ I E B ( x Ei , r Ei − α i ) , dn ! . The set E n = F ∪ i ∈ I E E n ( i ) ∪ X + ∪ X − is in C n (Γ , Γ , Ω). We do not prove that it is a (Γ n , Γ n )-cutsetsince the proof is very similar to the previous proofs. We control the capacity of the cutset E n inside the balls B ( x Ei , r Ei − α i ) , for i ∈ I E . Since E n ( i ) is a cutset corresponding to the event G n ( x Ei , r Ei − α i , v Ei , ε , f ( x Ei ) + η + M κ d √ ε ), it follows that V ( E n ( i )) α d − ( r Ei − α i ) d − n d − ≤ f ( x Ei ) + η + M κ d √ ε . If f ( x Ei ) < η , it is enough to conclude that (cid:12)(cid:12)(cid:12)(cid:12) V ( E n ( i )) α d − ( r Ei − α i ) d − n d − − f ( x Ei ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ η + M κ d √ ε occurs. Otherwise, let us definer( E n ( i )) = (cid:26) x ∈ Z dn ∩ B ( x Ei , ¯ r Ei , v Ei ) : there exists a path from x to ∂ − n B ( x Ei , ¯ r Ei , v Ei )in ( Z dn ∩ B ( x Ei , ¯ r Ei , v Ei ) , E dn \ E n ( i )) (cid:27) . Since E n ( i ) ∩ B ( x Ei , ¯ r Ei ) ⊂ cyl(disc( x Ei , ¯ r Ei , v Ei ) , ε r Ei ), we haver( E n ( i ))∆ B − ( x Ei , ¯ r Ei , v Ei ) ⊂ cyl(disc( x Ei , ¯ r Ei , v Ei ) , ε r Ei ) ∩ Z dn . It follows that for n large enoughcard(r( E n ( i ))∆ B − ( x Ei , ¯ r Ei , v Ei )) ≤ ε α d − ( r Ei ) d n d ≤ εα d (¯ r Ei ) d n d where we recall that ε = α d α d − − d − ε . (4.24)Hence, if V ( E n ( i ) ∩ B ( x Ei , ¯ r Ei )) ≤ (1 − η ) f ( x Ei ) α d − (¯ r Ei ) d − n d − then the event G n ( x Ei , ¯ r Ei , v Ei , ε, (1 − η ) f ( x Ei )) occurs. Consequently, on the event G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Ei , ¯ r Ei , v Ei , ε, (1 − η ) f ( x Ei )) c , we have V ( E n ( i )) ≥ (1 − η ) f ( x Ei ) α d − (¯ r Ei ) d − n d − and V ( E n ( i )) α d − ( r Ei − α i ) d − n d − ≥ (1 − η )(1 −√ ε ) d − f ( x Ei ) ≥ (1 − η )(1 − ( d − √ ε ) f ( x Ei ) ≥ f ( x Ei ) − ( η − ( d − √ ε ) M .
It follows that on the event E (3) ∩ E (4) ∀ i ∈ I E (cid:12)(cid:12)(cid:12)(cid:12) V ( E n ( i )) α d − ( r Ei − α i ) d − n d − − f ( x Ei ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ η (1 + M κ d ) . (4.25)where we choose ε small enough depending on η and d (we recall that κ d ≥ We prove that ( E n , µ n ( E n )) ∈ U . For i ∈ I E such that x Ei ∈ ∂ ∗ E ∩ Ω, we have by constructionthat R( E n ) ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ⊂ cyl(disc( x Ei , r Ei , v Ei ) , ε r Ei ). Using (4.14), we have L d ((R( E n ) ∩ B ( x Ei , r Ei ))∆( E ∩ B ( x Ei , r Ei ))) ≤ L d ((R( E n ) ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) + L d ( E ∩ B ( x Ei , r Ei ))∆ B − ( x Ei , r Ei , v Ei )) ≤ (4 ε α d − + εα d )( r Ei ) d .
42t is easy to check that for any i ∈ I E L d ((R( E n ) ∩ B ( x Ei , r Ei ))∆( E ∩ B ( x Ei , r Ei ))) ≤ (4 ε α d − + εα d )( r Ei ) d . It follows that for n large enough, we have using proposition 1.10 L d (R( E n )∆ E ) ≤ L d ( V ( E , d/n )) + X i ∈ I E L d ((R( E n ) ∩ B ( x Ei , r Ei ))∆( E ∩ B ( x Ei , r Ei ))) ≤ dn H d − ( E ) + X i ∈ I E (4 ε α d − + εα d )( r Ei ) d ≤ ε X i ∈ I E α d ( r Ei ) d ≤ ε L d ( V (Ω , ε was defined in (4.24).Then, we choose ε small enough depending on Ω, d and ξ such that2 ε L d ( V (Ω , ≤ ξ . (4.26)Let us compute the cardinal of the set F . By proposition 1.10 and inequality (4.12) and (4.13) and bythe same computations as in (3.32), for n large enoughcard( F ) ≤ d H d − (cid:0) E \ ∪ i ∈ I E B ( x Ei , r Ei − α i ) (cid:1) n d − ≤ d (cid:0) d √ ε H d − ( E ) + ε (cid:1) n d − ≤ η n d − (4.27)for ε small enough depending on η , d and H d − ( E ). Let j ∈ { , . . . , p } . Using (4.1) and the fact that r Ei ≤ δ for any i ∈ I E , we have by similar computations as in (3.31), using (4.12) and (4.13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ( g j ) − X i ∈ I E ν ( B ( x Ei , r Ei − α i )) g j ( x Ei ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k g j k ∞ ν (cid:0) E \ ∪ i ∈ I E B ( x Ei , r Ei − α i ) (cid:1) + X i ∈ I E Z B ( x Ei ,r Ei − α i ) ∩ E | g j ( x ) − g j ( x Ei ) | f ( x ) d H d − ( x ) ≤ k g j k ∞ M H d − ( E \ ∪ i ∈ I E B ( x Ei , r Ei − α i )) + ε Z E f ( x ) d H d − ( x ) ≤ k g j k ∞ M H d − ( E \ ∪ i ∈ I E B ( x Ei , r Ei )) + X i ∈ I E (cid:0) H d − ( B ( x Ei , r Ei ) ∩ E ) − H d − ( B ( x Ei , ¯ r Ei )) ∩ E ) (cid:1)! + εν (Ω) ≤ k g j k ∞ M ( ε + 2 d √ ε H d − ( E )) + εν (Ω) . (4.28)Similarly, we have using (4.27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ n ( g j ) − X i ∈ I E µ n ( B ( x Ei , r Ei − α i )) g j ( x Ei ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k g j k ∞ µ n (cid:0) Ω \ ∪ i ∈ I E B ( x Ei , r Ei − α i ) (cid:1) + X i ∈ I E (cid:16) µ n ( g j B ( x Ei ,r Ei − α i ) ) − µ n ( B ( x Ei , r Ei − α i )) g j ( x Ei ) (cid:17) ≤ M k g j k ∞ n d − (card( F ) + card( X + ∪ X − )) + εµ n (Ω) . (4.29)Let i ∈ I E . Using (4.19) and (4.25), we have (cid:12)(cid:12) µ n ( B ( x Ei , r Ei − α i )) − ν ( B ( x Ei , r Ei − α i )) (cid:12)(cid:12) ≤ (cid:12)(cid:12) µ n ( B ( x Ei , r Ei − α i )) − f ( x Ei ) α d − ( r Ei − α i ) d − (cid:12)(cid:12) + (cid:12)(cid:12) f ( x Ei ) α d − ( r Ei − α i ) d − − ν ( B ( x Ei , r Ei − α i )) (cid:12)(cid:12) ≤ ( η (1 + M κ d ) + ε ) α d − ( r Ei − α i ) d − .
43t follows using (4.13) that X i ∈ I E (cid:12)(cid:12) µ n ( B ( x Ei , r Ei − α i )) g j ( x Ei ) − ν ( B ( x Ei , r Ei − α i )) g j ( x Ei ) (cid:12)(cid:12) ≤ η (1 + M κ d ) H d − ( E ) k g j k ∞ (4.30)where we recall that ε ≤ η . Combining inequalities (4.23), (4.27), (4.28), (4.29) and (4.30), we have | µ n ( g j ) − ν ( g j ) | ≤ C η (4.31)where C depends on d , E , g j and M . For η ≤ ξ /C , we have | µ n ( g j ) − ν ( g j ) | ≤ ξ . Finally, we have( E n , µ n ) ∈ U ⊂ U . We prove that E n ∈ C n ( u ( η )) . Let G n ∈ C n (Γ , Γ , Ω). If V ( G n ) ≥ d M H d − (Γ ) n d − then V ( G n ) ≥ V ( E n ). If V ( G n ) ≤ d M H d − (Γ ) n d − , then on the event E (0) , we have card( G n ) ≤ βn d − and R( G n ) ∈ C β . Let F ∈ A such that L d (R( G n )∆ F ) ≤ ε F . Let i ∈ I F such that x Fi ∈ F . Thanks tothe choice of ε F and inequality (4.5), when x Fi ∈ Ω and B ( x Fi , r Fi ) ⊂ Ω, we can prove that if V ( G n ∩ B ( x Fi , r Fi )) ≤ (1 − η ) ν G ( v Fi ) α d − ( r Fi ) d − n d − , then, the event G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi )) occurs. By inequalities (4.6) and (4.7), when x Fi / ∈ Ω,we can prove as in section 5.2 in [2] that the same result holds. Similarly, for x Fi ∈ F ∩ E such that f ( x Fi ) ≥ η , if V ( G n ∩ B ( x Fi , r Fi )) ≤ (1 − η ) f ( x Fi ) α d − ( r Fi ) d − n d − then the event G n ( x Fi , r Fi , v Fi , ε, (1 − η ) f ( x Fi )) occurs. Hence, on the event E (1) ∩ E (2) , using the factthat the balls are disjoint, we have V ( G n ) n d − ≥ X i ∈ I F V ( G n ∩ B ( x Fi , r Fi )) n d − ≥ (1 − η ) X i ∈ I F : x Fi ∈ F \ E ν G ( v Fi ) α d − ( r Fi ) d − + (1 − η ) X i ∈ I F : x Fi ∈ F ∩ E , f ( x Fi ) ≥ η f ( x Fi ) α d − ( r Fi ) d − . Besides, we have using inequalities (4.3), (4.4) and (4.9) X i ∈ I F : x Fi ∈ F ∩ E , f ( x Fi ) ≥ η f ( x Fi ) α d − ( r Fi ) d − = X i ∈ I F : x Fi ∈ F ∩ E f ( x Fi ) α d − ( r Fi ) d − − X i ∈ I F : x Fi ∈ F ∩ E , f ( x Fi ) <η f ( x Fi ) α d − ( r Fi ) d − ≥ X i ∈ I F : x Fi ∈ F ∩ E Z B ( x Fi ,r Fi ) ∩ F f ( x ) d H d − ( x ) − εα d − ( r Fi ) d − ! − η X i ∈ I F : x Fi ∈ F ∩ E , α d − ( r Fi ) d − ≥ Z F ∩ E f ( x ) d H d − ( x ) − εM − ε + η ) H d − ( F ∩ E ) ≥ Z F ∩ E f ( x ) d H d − ( x ) − η ( M + 4( β + H d − ( ∂ Ω))44here we recall that ε ≤ η and E, F ∈ C β . Using inequalities (4.2), (4.3), (4.4) and (4.8) we have X i ∈ I F : x Fi ∈ F \ E ν G ( v Fi ) α d − ( r Fi ) d − ≥ X i ∈ I F : x Fi ∈ F \ E Z F ∩ B ( x Fi ,r Fi ) ν G ( n • ( y )) d H d − ( y ) − εα d − ( r Fi ) d − ! ≥ Z F \V ( E ,δ F ) ν G ( n • ( y )) d H d − ( y ) − M H d − ( F \ V ( E , δ F )) \ [ i ∈ I F B ( x Fi , r Fi ) ! − ε H d − ( F ) ≥ Z F \V ( E ,δ F ) ν G ( n • ( y )) d H d − ( y ) − η ( M + β + H d − ( ∂ Ω)) ≥ Z ( F \ E ) ν G ( n • ( y )) d H d − ( y ) − ε − η ( M + β + H d − ( ∂ Ω))where we use in the second last inequality that F ∈ C β . Combining the three previous inequalities, usingthe fact that ( E, f ) is minimal, we obtain V ( G n ) n d − ≥ (1 − η ) Cap( E, f ) − η (3 M + 6 β + 6 H d − ( ∂ Ω)) . Note that Cap(
E, f ) ≤ dM H d − (Γ ). It follows that V ( G n ) n d − ≥ Cap(
E, f ) − η (3 M + 6 β + 6 H d − ( ∂ ∗ Ω) − dM H d − (Γ )) . (4.32)Besides, on the event E (4) , using (4.19), (4.23), (4.25) and (4.27), we have V ( E n ) n d − ≤ X i ∈ I E V ( E n ( i )) n d − + Mn d − (card( F ) + card( X + ∪ X − )) ≤ X i ∈ I E ( f ( x Ei ) + η (1 + M κ d )) α d − ( r Ei ) d − + 2 ηM ≤ Cap(
E, f ) + X i ∈ I E ( ε + η (1 + M κ d )) α d − ( r Ei ) d − + 2 M η ≤ Cap(
E, f ) + η (3 + M κ d ) H d − ( E ) + 2 M η ≤ Cap(
E, f ) + η (cid:0) (3 + M κ d )( β + H d − ( ∂ Ω)) + 2 M (cid:1) . (4.33)Finally, combining inequalities (4.32) and (4.33), there exists a function u (depending on M , β , Ω) suchthat lim η → u ( η ) = 0 and on the event E (1) ∩ E (2) ∩ E (4) , for any G n ∈ C n (Γ , Γ , Ω) V ( G n ) ≥ V ( E n ) − u ( η ) n d − . It follows that E n ∈ C n ( u ( η )). The claim is thus proved. Step 4. We conclude by estimating the probabilities of the events.
We have P (cid:16) E (4) ∩ E (1) (cid:17) = P ( E (2) ∩ E (3) ∩ E (4) ∩ E (1) ) + P (cid:16) (( E (2) ) c ∩ E (4) ∩ E (1) ) ∪ (( E (3) ) c ∩ E (4) ∩ E (1) ) (cid:17) . (4.34)45oreover, we have P (cid:16) (( E (2) ) c ∩ E (4) ∩ E (1) ) ∪ (( E (3) ) c ∩ E (4) ∩ E (1) ) (cid:17) ≤ P (( E (2) ) c ∩ E (4) ) + P (( E (3) ) c ∩ E (4) ) ≤ X F ∈A X j ∈ I F : x Fj ∈ F ∩ E ,f ( x Fj ) ≥ η P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj )) ! + X j ∈ I E : f ( x Ej ) ≥ η P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej )) ! . (4.35)Let us estimate P (( E (3) ) c ∩ E (4) ). Let j ∈ I E such that f ( x Ej ) ≥ η , since the balls are disjoint, we haveusing the independencelog P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε, f ( x Ei ) + η ) ∩ G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej )) ! − log P ( E (4) ) ≤ log P \ i ∈ I E : i = j G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej )) − log P ( E (4) )= log P ( G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej ))) − log P ( G n ( x Ej , ¯ r Ej , v Ej , ε , f ( x Ej ) + η )) . (4.36)Let κ > n →∞ n d − log P ( G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej ))) ≤ − g ( ε ) α d − (¯ r Ej ) d − J v Ej (1 − η ) f ( x Ej ) + κ √ εg ( ε ) ! . (4.37)Let us choose ε (and so ε ) small enough depending as δ in (2.7) (see lemma 2.6). Using lemma 2.6, wehave lim inf n →∞ n d − log P ( G n ( x Ej , ¯ r Ej , v Ej , ε , f ( x Ej ) + η )) ≥ − α d − (¯ r Ej ) d − J v Ej ( f ( x Ej )) . (4.38)Pick ε small enough such that 1 g ( ε ) ≤ η and 2 κ √ ε ≤ η , (4.39)(remember that g ( ε ) goes to 0 when ε goes to 0). We recall that f ( x Ej ) ≥ η . Using that the map J v Ej isnon increasing, we get J v Ej (1 − η ) f ( x Ej ) + κ √ εg ( ε ) ! ≥ J v Ej (cid:0) f ( x Ej ) + 2 κ √ ε − η f ( x Ej ) (cid:1) ≥ J v Ej (cid:18) f ( x Ej ) − η (cid:19) . (4.40)Using that J v Ej ( ν G ( v Ej )) = 0 and the convexity of the map J v Ej , we obtain J v Ej ( f ( x Ej )) ≤ λ J v Ej (cid:18) f ( x Ej ) − η (cid:19) (4.41)where λ ∈ ]0 ,
1[ is chosen such that f ( x Ej ) = λ (cid:18) f ( x Ej ) − η (cid:19) + (1 − λ ) ν G ( v Ej ) .
46e recall that f ( x Ej ) ≤ ν G ( v Ej ) since ( E, f ) ∈ T . If f ( x Ej ) = ν G ( v Ej ) then inequality (4.41) is trivial.Otherwise, such a λ ∈ ]0 ,
1[ exists. Choose ε small enough depending on η and ν G such thatsup v ∈ S d − ν G ( v )sup v ∈ S d − ν G ( v ) + η / < g ( ε ) . Using that the map x x/ ( x + η /
2) is non-decreasing on R + , it follows that λ = ν G ( v Ej ) − f ( x Ej ) ν G ( v Ej ) − f ( x Ej ) + η / ≤ sup v ∈ S d − ν G ( v )sup v ∈ S d − ν G ( v ) + η / < g ( ε ) . (4.42)Hence, we have λ < g ( ε ). Combining inequalities (4.36), (4.37), (4.38), (4.40), (4.41) and (4.42), weobtain thatlim sup n →∞ n d − log P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej )) ! − log P ( E (4) ) ! ≤ − g ( ε ) α d − (¯ r Ej ) d − J v Ej (cid:18) f ( x Ej ) − η (cid:19) + λα d − (¯ r Ej ) d − J v Ej (cid:18) f ( x Ej ) − η (cid:19) < . (4.43)Let us estimate P (( E (2) ) c ∩ E (4) ). Let F ∈ A and j ∈ I F such that x Fj ∈ F ∩ E and f ( x Fj ) ≥ η . We have P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj )) ! ≤ P \ i ∈ I E : B ( x Ei , ¯ r Ei ) ∩ B ( x Fj ,r Fj )= ∅ G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj )) . Using the independence and the previous inequality, it follows thatlog P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj )) ! − log P ( E (4) ) ≤ log P ( G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj ))) − log P \ i ∈ I E : B ( x Ei , ¯ r Ei ) ∩ B ( x Fj ,r Fj ) = ∅ G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) (4.44)and thanks to lemma 2.5 and the choice of ε as in (4.39), we havelim sup n →∞ n d − log P ( G n ( x Fj , r Fj , v Fj , ε, (1 − η ) f ( x Fj ))) ≤ − g ( ε ) α d − ( r Fj ) d − J v Fj (cid:18) f ( x Fj ) − η (cid:19) . (4.45)Besides, we have using the independence, lemma 2.6 and inequalities (4.4), (4.10), (4.13) and (4.20)lim inf n →∞ n d − log P \ i ∈ I E : B ( x Ei , ¯ r Ei ) ∩ B ( x Fj ,r Fj ) = ∅ G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ≥ − X i ∈ I E : B ( x Ei , ¯ r Ei ) ∩ B ( x Fj ,r Fj ) = ∅ α d − (¯ r Ei ) d − J v Ei ( f ( x Ei )) ≥ − X i ∈ I E : B ( x Ei , ¯ r Ei ) ∩ B ( x Fj ,r Fj ) = ∅ Z B ( x Ei , ¯ r Ei ) ∩ E J n • ( x ) ( f ( x )) d H d − ( x ) + εα d − (¯ r Ei ) d − ! − Z B ( x Fj ,r Fj ) ∩ E J n • ( x ) ( f ( x )) d H d − ( x ) − Z E ∩ B ( x Fj ,r Fj + δ ) \ B ( x Fj ,r Fj ) J n • ( x ) ( f ( x )) d H d − ( x ) − ε (cid:0) H d − ( E ∩ B ( x Fi , r Fi )) + H d − ( E ∩ ( B ( x Fi , r Fi + δ ) \ B ( x Fi , r Fi )) (cid:1) ≥ − α d − ( r Fj ) d − J v Fj ( f ( x Fj )) − εα d − ( r Fj ) d − − ε (cid:0) α d − ( r Fj ) d − + 2 εα d − ( r Fj ) d − (cid:1) ≥ − α d − ( r Fj ) d − J v Fj ( f ( x Fj )) − εα d − ( r Fj ) d − (4.46)where we use that r Ei ≤ δ and that δ was chosen such that it satisfies (4). We can choose ε smallenough depending on η such that7 ε < inf v ∈ S d − J v (cid:18) ν G ( v ) − η (cid:19) (cid:18) g ( ε ) − sup v ∈ S d − ν G ( v )sup v ∈ S d − ν G ( v ) + η / (cid:19) . (4.47)We recall that g ( ε ) goes to 1 when ε goes to 0. Hence, when ε goes to 0, the limit of right hand side ispositive. Using the convexity as above, we obtain J v Fj ( f ( x Fj )) ≤ λ J v Fj (cid:18) f ( x Fj ) − η (cid:19) (4.48)where λ ≤ sup v ∈ S d − ν G ( v )sup v ∈ S d − ν G ( v ) + η / . Combining inequalities (4.44), (4.45), (4.46) and (4.48), we get thanks to the choice of ε as in (4.47)lim sup n →∞ n d − log P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ∩ G n ( x Ej , ¯ r Ej , v Ej , ε, (1 − η ) f ( x Ej )) ! / P ( E (4) ) ! ≤ − g ( ε ) α d − ( r Fj ) d − J v Fj (cid:18) f ( x Fj ) − η (cid:19) + λα d − ( r Fj ) d − J v Fj (cid:18) f ( x Ej ) − η (cid:19) + 7 εα d − ( r Fj ) d − < . (4.49)Combining inequalities (4.35), (4.43), (4.49) with lemma 1.11 thatlim sup n →∞ n d − log P (( E (2) ∩ E (3) ) c ∩ E (4) ∩ E (1) ) < lim inf n →∞ n d − log P ( E (4) ) . (4.50)Let us now estimate P ( E (1) ). Using FKG inequality, we have P (cid:16) E (1) (cid:17) ≥ Y F ∈A Y i ∈ I F : x Fi ∈ F \ E P ( G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi )) c ) . Let F ∈ A and i ∈ I F . Thanks to the choice of ε in (4.39) and lemma 2.5, we havelim sup n →∞ n d − log P ( G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi ))) < − g ( ε ) α d − ( r Fi ) d − J v Fi (cid:18) ν G ( v Fi ) − η (cid:19) < . It follows by lemma 1.11 thatlim inf n →∞ n d − log P ( G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi )) c ) = 0andlim inf n →∞ n d − log P (cid:16) E (1) (cid:17) ≥ X F ∈A X i ∈ I F : x Fi ∈ F \ E lim inf n →∞ n d − log P ( G n ( x Fi , r Fi , v Fi , ε, (1 − η ) ν G ( v Fi )) c ) = 0 .
48t follows that lim inf n →∞ n d − log P (cid:16) E (1) (cid:17) = 0 . (4.51)Using that E (1) and E (4) are independent equalities (4.34) and (4.51), inequality (4.50) and lemma 1.11lim inf n →∞ n d − log P (cid:16) E (4) (cid:17) ≤ lim inf n →∞ n d − log P (cid:16) E (4) ∩ E (1) (cid:17) = lim inf n →∞ n d − log P ( E (1) ∩ E (2) ∩ E (3) ∩ E (4) ) . (4.52)Using (4.52), lemma 2.6, inequalities (4.13) and (4.20), we havelim inf n →∞ n d − log P ( E (1) ∩ E (2) ∩ E (3) ∩ E (4) ) ≥ lim inf n →∞ n d − log P ( E (4) ) ≥ lim inf n →∞ n d − log P \ i ∈ I E G n ( x Ei , ¯ r Ei , v Ei , ε , f ( x Ei ) + η ) ! ≥ − X i ∈ I E α d − ¯ r Ei J v Ei ( f ( x Ei )) ≥ − X i ∈ I E Z E ∩ B ( x Ei ,r Ei ) J n • ( y ) ( f ( y )) d H d − ( y ) + εα d − ( r Ei ) d − ! ≥ − e I ( E, ν ) − ε H d − ( E ) . Besides, we have up to choosing a smaller ε lim sup n →∞ n d − log P (( E (0) ) c ) ≤ − e I ( E, ν ) < − e I ( E, ν ) − ε H d − ( E ) . By lemma 1.11, it follows that for small enough ε lim inf n →∞ n d − log P ( E (0) ∩ E (1) ∩ E (2) ∩ E (3) ∩ E (4) ) = lim inf n →∞ n d − log P ( E (1) ∩ E (2) ∩ E (3) ∩ E (4) ) . For any η > ε small enough depending on η , we havelim inf n →∞ n d − log P ( ∃E n ∈ C n ( u ( η )) : (R( E n ) , µ n ( E n )) ∈ U ) ≥ lim inf n →∞ n d − log P ( E (0) ∩ E (1) ∩ E (2) ∩ E (3) ∩ E (4) ) ≥ − e I ( E, ν ) − ε H d − ( E ) . By letting first ε go to 0 and then η go to 0 we obtain the expected result. Calibration of constants.
We explain here in which order the constants are chosen. We first choose η such that it satisfies η ≤ ξ C where the constant C was defined in (4.31). Then we choose ε small enough depending on η , d , Ω and ξ such that it satisfies (4.26), (4.39) and (4.47). Once ε is fixed, we can define δ as in (4.1), δ F for each F ∈ C β as in (4.2), the covering of C β . Once the covering of C β is chosen, we can define δ as in (4).Finally, we can build a covering of E depending on ε and on the covering of C β . The aim of this section is to prove the following result.49 roposition 5.1.
Let ( E, f ) ∈ T . Write ν = f H d − | E . If I ( E, f ) < ∞ , then for any δ ∈ ]0 , , thereexists a neighborhood U of ( E, ν ) such that lim sup n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ − (1 − δ ) I ( E, f ) . If I ( E, f ) = + ∞ , then for any t > there exists a neighborhood U of ( E, ν ) such that lim sup n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ − t . Before proving this result, let us introduce some properties of the rate function J v that will be usefulin the proof of proposition 5.1. J v We already know that for a fixed v ∈ S d − , the map λ
7→ J v ( λ ) is lower semi-continuous. We aimto prove in this section that the map ( λ, v )
7→ J v ( λ ) is also lower semi-continuous. The rate function J satisfies the weak triangle inequality in the following sense. Proposition 5.2 (weak triangle inequality for J ) . Let ( ABC ) be a non-degenerate triangle in R d andlet v A , v B , v C be the exterior normal unit vectors to the sides [ BC ] , [ AC ] , [ AB ] in the plane spanned by A, B, C . Then, for any λ, µ ≥ H ([ BC ]) J v A (cid:18) λ H ([ AC ]) + µ H ([ AB ]) H ([ BC ]) (cid:19) ≤ H ([ AC ]) J v B ( λ ) + H ([ AB ]) J v C ( µ ) . Proof.
The proof is an adaptation of the proof of Proposition 11.6 in [1]. This result was proved forthe dimension 2 in lemma 3.1. in [11]. We only treat the case where the triangle (
ABC ) is such that −−→ BA · −−→ BC ≥ −→ CA · −−→ CB ≥ e , . . . , e d ) be an orthonormal basis such that e and e belong to the spacespanned by A, B, C . Let ε, h be such that 0 < ε ≤ ≤ h and λ, µ ≥
0. Let K be the compact convexset defined by K = ( x + d X i =3 u i e i : x ∈ ( ABC ) , ( u , . . . , u d ) ∈ [0 , h ] d − ) . The boundary of K consists of the three following hyperrectangles R A = ( x + d X i =3 u i e i : x ∈ [ BC ] , ( u , . . . , u d ) ∈ [0 , h ] d − ) ,R B = ( x + d X i =3 u i e i : x ∈ [ AC ] , ( u , . . . , u d ) ∈ [0 , h ] d − ) ,R C = ( x + d X i =3 u i e i : x ∈ [ AB ] , ( u , . . . , u d ) ∈ [0 , h ] d − ) and the set T = [ ≤ j ≤ d ( x + d X i =3 u i e i : x ∈ ( ABC ) , u j ∈ { , h } , ( u , . . . , u j − , u j +1 , . . . , u d ) ∈ [0 , h ] d − ) . We can define R εB and R εC such that (see figure 6): • R εB ⊂ R B , R εC ⊂ R C and cyl( R εB , ε ) ∩ cyl( R εC , ε ) = ∅ , • for h ≥ R εB , ε ) ∪ cyl( R εC , ε ) ⊂ cyl( R A , h ), • we have for any h >
0, lim ε → H d − ( R B \ R εB ) = 0 and lim ε → H d − ( R C \ R εC ) = 0.50igure 6 – Representation of R εB and R εC .We denote by E εn the edges in E dn that have at least one endpoint in V ( R B \ R εB , d/n ) ∪ V ( R C \ R εC , d/n ) ∪ V ( T, d/n ) . Using proposition 1.10, there exists a positive constant c d depending only on d such thatcard(E εn ) ≤ c d (cid:0) H d − ( R B \ R εB ) + H d − ( R C \ R εC ) + H d − ( T ) (cid:1) n d − ≤ c d (cid:0) H d − ( R B \ R εB ) + H d − ( R C \ R εC ) + 2( d − H ( ABC ) h d − (cid:1) n d − . Set E = { τ n ( R εB , ε ) ≤ λ H d − ( R εB ) n d − } ∩ { τ n ( R εC , ε ) ≤ µ H d − ( R εC ) n d − } . Note that if E εB (respectively E εC ) is a cutset for τ n ( R B , ε ) (resp. τ n ( R C , ε )), then E εB ∪ E εC ∪ E εn is acutset for τ n ( R A , h ). Hence, we have E ⊂ { τ n ( R A , h ) ≤ ( λ H d − ( R B ) + µ H d − ( R C )) n d − + card(E εn ) M } . Hence, we have using the independence P (cid:18) τ n ( R A , h ) n d − ≤ λ H d − ( R B ) + µ H d − ( R C ) + c d (cid:0) f ( ε ) + 2( d − H ( ABC ) h d − (cid:1) M (cid:19) ≥ P (cid:0) τ n ( R A , h ) ≤ ( λ H d − ( R B ) + µ H d − ( R C )) n d − + card(E εn ) M (cid:1) ≥ P ( τ n ( R εB , ε ) ≤ λ H d − ( R εB ) n d − ) P ( τ n ( R εC , ε ) ≤ µ H d − ( R εC ) n d − )where f ( ε ) = H d − ( R B \ R εB ) + H d − ( R C \ R εC ). Using theorem 1.13, we have H d − ( R A ) J v A λ H d − ( R B ) + µ H d − ( R C ) + c d (cid:0) f ( ε ) + 2( d − H ( ABC ) h d − (cid:1) M ) H d − ( R A ) ! ≤ H d − ( R εB ) J v B ( λ ) + H d − ( R εC ) J v C ( µ ) .
51e recall that H d − ( R A ) = h d − H ([ BC ]), H d − ( R B ) = h d − H ([ AC ]) and H d − ( R C ) = h d − H ([ AB ]).We also recall that J v A is lower semi-continuous. By letting ε goes to 0 we obtain H d − ( R A ) J v A (cid:18) λ H ([ AC ]) + µ H ([ AB ]) + 2 c d ( d − H ( ABC ) h − M H ([ BC ]) (cid:19) ≤ lim inf ε → H d − ( R A ) J v A λ H d − ( R B ) + µ H d − ( R C ) + c d (cid:0) f ( ε ) + 2( d − H ( ABC ) h d − (cid:1) M ) H d − ( R A ) ! ≤ H d − ( R B ) J v B ( λ ) + H d − ( R C ) J v C ( µ ) . By dividing the inequality by h d − , we obtain H ([ BC ]) J v A (cid:18) λ H ([ AC ]) + µ H ([ AB ]) + 2 c d ( d − H ( ABC ) h − ) M ) H ([ BC ]) (cid:19) ≤ H ([ AC ]) J v B ( λ ) + H ([ AB ]) J v C ( µ ) . Letting h go to infinity, using again the fact that J v A is lower semi-continuous, yields the result: H ([ BC ]) J v A (cid:18) λ H ([ AC ]) + µ H ([ AB ]) H ([ BC ]) (cid:19) ≤ H ([ AC ]) J v B ( λ ) + H ([ AB ]) J v C ( µ ) . We can deduce from proposition 5.2 the lower semi-continuity of J . Corollary 5.3 (The function J is lower semi-continuous) . For any λ ≥ , v ∈ S d − , for any sequence ( λ n ) n ≥ of positive real numbers that converges towards λ , for any sequence ( v n ) n ≥ of S d − that con-verges towards v , we have lim inf n →∞ J v n ( λ n ) ≥ J v ( λ ) . Proof.
Let λ ≥
0. Let v ∈ S d − . Let ( λ n ) n ≥ be a sequence of positive real numbers that convergestowards λ . Let ( v n ) n ≥ be a sequence of S d − that converges towards v . Let P n be the plan spanned by( O, v, v n ). Let A and A n be points of R d in P n such that k−→ OA k = k−−→ OA n k = 1, −→ OA · v = −−→ OA n · v n = 0.Note that k−−→ AA n k = k−−→ OA n − −→ OA k = k v n − v k where we use that there is an isometry (rotation of π/ P n ) that sends −→ OA on v and −−→ OA n on v n and such that −−→ OA n · −→ OA ≥
0. It follows thatlim n → H ([ AA n ]) = 0 . Let v n be the exterior normal unit vector to the side [ AA n ] of the triangle OAA n in P n (see figure 7).Figure 7 – The triangle OAA n in the plane P n Applying proposition 5.2, we have H ([ OA ]) J v (cid:18) H ([ OA n ]) λ n + H ([ AA n ]) ν G ( v n ) H ([ OA ]) (cid:19) ≤ H ([ AA n ]) J v n ( ν G ( v n )) + H ([ OA n ]) J v n ( λ n )52nd it yields that J v ( λ n + H ([ AA n ]) ν G ( v n )) ≤ H ([ AA n ]) J v n ( ν G ( v n )) + J v n ( λ n ) = J v n ( λ n ) . Besides, we have λ n + H ([ AA n ]) inf v ∈ S d − ν G ( v ) ≤ λ n + H ([ AA n ]) ν G ( v n ) ≤ λ n + H ([ AA n ]) sup v ∈ S d − ν G ( v ) . As a result, we have lim n →∞ λ n + H ([ AA n ]) ν G ( v n ) = λ . Since the function α
7→ J v ( α ) is lower semi-continuous, we have J v ( λ ) ≤ lim inf n →∞ J v ( λ n + H ([ AA n ]) ν G ( v n )) ≤ lim inf n →∞ J v n ( λ n ) . The result follows.
This section corresponds to the step 3 of the sketch of the proof in the introduction. For (
E, f ) ∈ T , wecover E by small disjoint balls. Then we prove that we can build a neighborhood U of ( E, ν ) adapted tothis covering such that we can upperbound the probability P ( ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U )using estimates on lower large deviations for the maximal flow in a ball. Proof of proposition 5.1.
Let ε >
0. By proposition 2.3, there exists a finite family of disjoint closed balls( B ( x i , r i , v i )) i ∈ I ∪ I ∪ I such that for i ∈ I , we have x i ∈ ∂ ∗ E ∩ Ω, for i ∈ I , we have x i ∈ ∂ ∗ Ω ∩ (Γ \ ∂ ∗ E )and for i ∈ I , we have x i ∈ ∂ ∗ E ∩ Γ and the following properties hold: H d − ( E \ ∪ i ∈ I ∪ I ∪ I B ( x i , r i ))) ≤ ε , (5.1) ∀ i ∈ I ∪ I ∪ I ∀ < r ≤ r i (cid:12)(cid:12)(cid:12)(cid:12) α d − r d − H d − ( E ∩ B ( x i , r )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (5.2) ∀ i ∈ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H d − ( ∂ ∗ E ∩ B ( x i , r i )) Z ∂ ∗ E ∩ B ( x i ,r i ) n E ( x ) d H d − ( x ) − n E ( x i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε , (5.3) ∀ i ∈ I ∪ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H d − ( ∂ ∗ Ω ∩ B ( x i , r i )) Z ∂ ∗ Ω ∩ B ( x i ,r i ) n Ω ( x ) d H d − ( x ) − n Ω ( x i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε , (5.4) ∀ i ∈ I ∪ I ∪ I (cid:12)(cid:12)(cid:12)(cid:12) H d − ( E ∩ B ( x i , r i )) ν ( B ( x i , r i )) − f ( x i ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , (5.5) ∀ i ∈ I B ( x i , r i ) ⊂ Ω and L d (( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (5.6) ∀ i ∈ I d ( B ( x i , r i ) , Γ ∪ E ) > L d ((Ω ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di , (5.7) ∀ i ∈ I d ( B ( x i , r i ) , Γ ) > L d (( E ∩ B ( x i , r i ))∆ B − ( x i , r i , v i )) ≤ εα d r di . (5.8)Let i ∈ I ∪ I ∪ I . Using inequality (5.2), we have ν ( B ( x i , r i (1 + ε )) \ B ( x i , r i )) = Z ( B ( x i ,r i (1+ ε )) \ B ( x i ,r i )) ∩ E f ( y ) d H d − ( y ) ≤ M (cid:0) H d − ( E ∩ B ( x i , r i (1 + ε ))) − H d − ( E ∩ B ( x i , r i )) (cid:1) ≤ M (cid:0) (1 + ε ) d − (1 − ε ) (cid:1) α d − r d − i ≤ M (1 + (2 d − ε − ε ) α d − r d − i ≤ M d εα d − r d − i . (5.9)53et f i be a continuous function taking its values in [0 ,
1] with support included in B ( x i , r i (1 + ε )) andsuch that f i = 1 on B ( x i , r i ). Using inequalities (5.2) and (5.5), we have ν ( f i ) ≤ ν ( B ( x i , r i )) + ν ( B ( x i , r i (1 + ε )) \ B ( x i , r i )) ≤ ( f ( x i ) + ε ) H d − ( E ∩ B ( x i , r i )) + 2 d M εα d − r d − i ≤ (( f ( x i ) + ε )(1 + ε ) + 2 d M ε ) α d − r d − i ≤ ( f ( x i ) + ε (2 + 2 d M + M )) α d − r d − i . We set ε E = εα d min i ∈ I ∪ I ∪ I r di . Let U be the weak neighborhood of ( E, ν ) in B ( R d ) × M ( R d ) defined by U = { F ∈ B ( R d ) : L d ( E ∆ F ) ≤ ε E } × (cid:8) ρ ∈ M ( R d ) : ∀ i ∈ I ∪ I ∪ I | ρ ( f i ) − ν ( f i ) | ≤ εα d − r d − i (cid:9) . (5.10)On the event {∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U } , we pick E n ∈ C n (Γ , Γ , Ω) such that(R( E n ) , µ n ( E n )) ∈ U .
If there are several possible choices, we choose according to a deterministic rule.We write µ n = µ n ( E n ) and E n = R( E n ). Let i ∈ I ∪ I ∪ I . We have µ n ( B ( x i , r i )) ≤ µ n ( f i ) ≤ ν ( f i ) + εα d − r d − i ≤ ( f ( x i ) + (3 + 2 d M + M ) ε ) α d − r d − i . Moreover, using that L d ( E n ∆ E ) ≤ εα d r di , we can prove as in section 5.2. in [2] that the following eventoccurs \ i ∈ I ∪ I G n ( x i , r i , n E ( x i ) , ε, f ( x i ) + κε ) \ i ∈ I G n ( x i , r i , n Ω ( x i ) , ε, f ( x i ) + κε )where we recall that the event G n was defined in section 2.3.1 and where we set κ = 3 + 2 d M + M . Usingthat the balls are disjoint, we have by independence P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ Y i ∈ I ∪ I P ( G n ( x i , r i , n E ( x i ) , ε, f ( x i ) + κε ) Y i ∈ I P ( G n ( x i , r i , n Ω ( x i ) , ε, f ( x i ) + κε ) . Let κ > n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ − g ( ε ) X i ∈ I ∪ I α d − r d − i J n E ( x i ) (cid:18) f ( x i ) + κε + κ √ εg ( ε ) (cid:19) − g ( ε ) X i ∈ I α d − r d − i J n Ω ( x i ) (cid:18) f ( x i ) + κε + κ √ εg ( ε ) (cid:19) ≤ − g ( ε )1 + ε X i ∈ I ∪ I H d − ( ∂ ∗ E ∩ B ( x i , r i )) J n E ( x i ) (cid:18) f ( x i ) + κε + κ √ εg ( ε ) (cid:19) − g ( ε )1 + ε X i ∈ I H d − ( ∂ ∗ Ω ∩ B ( x i , r i )) J n Ω ( x i ) (cid:18) f ( x i ) + κε + κ √ εg ( ε ) (cid:19) where we use inequality (5.2) in the last inequality. Set E ε = ∪ i ∈ I ∪ I ∪ I B ( x i , r i ) ∩ ( ∂ ∗ E ∪ ∂ ∗ Ω). Forany x ∈ E ε , we denote by c ε ( x ) the unique x i ∈ I ∪ I ∪ I such that x ∈ B ( x i , r i ). It follows thatlim sup n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ − g ( ε )1 + ε Z E J n • ( c ε ( x )) (cid:18) f ( c ε ( x )) + κε + κ √ εg ( ε ) (cid:19) E ε ( x ) d H d − ( x )54here n • ( x ) = n Ω ( x ) for x ∈ ∂ ∗ Ω \ ∂ ∗ E and n • ( x ) = n E ( x ) for x ∈ ∂ ∗ E . We aim to study the limit ofthe right hand side when ε goes to 0 along a given sequence. For any i ∈ I ∪ I ∪ I , thanks to inequality(5.5), we have H d − (cid:0) { y ∈ B ( x i , r i ) ∩ E : | f ( y ) − f ( x i ) | ≥ √ ε } (cid:1) ≤ √ ε H d − ( B ( x i , r i ) ∩ E ) . Moreover, thanks to inequalities (5.3) and (5.4), we have for i ∈ I ∪ I H d − (cid:0) { y ∈ B ( x i , r i ) ∩ ∂ ∗ E : k n E ( y ) − n E ( x i ) k ≥ √ ε } (cid:1) ≤ √ ε H d − ( B ( x i , r i ) ∩ E )and for i ∈ I H d − (cid:0) { y ∈ B ( x i , r i ) ∩ ( ∂ ∗ E ∪ ∂ ∗ Ω) : k n Ω ( y ) − n Ω ( x i ) k ≥ √ ε } (cid:1) ≤ √ ε H d − ( B ( x i , r i ) ∩ E ) . Set N ε = [ i ∈ I ∪ I ∪ I (cid:8) y ∈ B ( x i , r i ) ∩ E : | f ( y ) − f ( x i ) | ≥ √ ε (cid:9) ∪ [ i ∈ I ∪ I (cid:8) y ∈ B ( x i , r i ) ∩ ∂ ∗ E : k n E ( y ) − n E ( x i ) k ≥ √ ε (cid:9) ∪ [ i ∈ I (cid:8) y ∈ B ( x i , r i ) ∩ ∂ ∗ Ω \ ∂ ∗ E : k n Ω ( y ) − n Ω ( x i ) k ≥ √ ε (cid:9) . Since the balls are disjoint, we have H d − ( N ε ) ≤ √ ε X i ∈ I ∪ I ∪ I H d − ( B ( x i , r i ) ∩ E ) ≤ √ ε H d − ( E ) ≤ √ ε ( P ( E, Ω) + H d − ( ∂ ∗ Ω)) . (5.11)For any p ≥
1, set ε p = p − . Let x ∈ lim inf p →∞ ( E ε p \ N ε p ) = ∪ p ≥ ∩ m ≥ p ( E ε m \ N ε m ). Hence, for p large enough, we have x ∈ ∩ m ≥ p ( E ε m \ N ε m ) and by construction for m ≥ p | f ( x ) − f ( c ε m ( x )) | ≤ √ ε m . It implies that lim m →∞ f ( c ε m ( x )) = f ( x ) . Similarly, if x ∈ ∂ ∗ E ∩ lim inf p →∞ ( E ε p \ N ε p ),lim p →∞ n E ( c ε p ( x )) = n E ( x )and if x ∈ ∂ ∗ Ω \ ∂ ∗ E ∩ lim inf p →∞ ( E ε p \ N ε p ),lim p →∞ n Ω ( c ε p ( x )) = n Ω ( x ) . Using inequality (5.1) and inequality (5.11), we have X p ≥ (cid:0) H d − ( N ε p ) + H d − ( E \ E ε p ) (cid:1) ≤ X p ≥ (cid:18) p ( P ( E, Ω) + H d − ( ∂ ∗ Ω)) + 1 p (cid:19) < ∞ . By Borel-Cantelli lemma, it yields H d − (cid:18) lim sup p →∞ N ε p ∪ ( E \ E ε p ) (cid:19) = 0 . Hence, we get H d − (cid:18) E ∩ (lim sup p →∞ N ε p ∪ ( E \ E ε p )) c (cid:19) = H d − ( E )55nd lim inf p →∞ E εp \N εp = E H d − -almost everywhere . (5.12)We havelim sup ε → g ( ε )1 + ε Z E J n • ( c ε ( x )) (cid:18) f ( c ε ( x )) + κε + κ √ εg ( ε ) (cid:19) E ε ( x ) d H d − ( x ) ≥ lim sup p →∞ g ( ε p )1 + ε p Z E J n • ( c εp ( x )) (cid:18) f ( c ε p ( x )) + κε p + κ √ ε p g ( ε p ) (cid:19) E εp \N εp ( x ) d H d − ( x ) ≥ lim inf p →∞ g ( ε p )1 + ε p Z E J n • ( c εp ( x )) (cid:18) f ( c ε p ( x )) + κε p + κ √ ε p g ( ε p ) (cid:19) E εp \N εp ( x ) d H d − ( x ) ≥ Z E lim inf p →∞ J n • ( c εp ( x )) (cid:18) f ( c ε p ( x )) + κε p + κ √ ε p g ( ε p ) (cid:19) E εp \N εp ( x ) d H d − ( x )where we use Fatou lemma in the last inequality. Using corollary 5.3 and (5.12), we have for H d − almostevery x ∈ E lim inf p →∞ J n • ( c εp ( x )) (cid:18) f ( c ε p ( x )) + κε p + κ √ ε p g ( ε p ) (cid:19) ≥ J n • ( x ) ( f ( x )) . Finally, it follows thatlim sup ε → g ( ε )1 + ε Z E J n • ( c ε ( x )) (cid:18) f ( c ε ( x )) + κε + κ √ εg ( ε ) (cid:19) E ε ( x ) d H d − ( x ) ≥ I ( E, f ) . Calibration of constants.
Let δ >
0. If I ( E, f ) < ∞ , we can choose ε such that g ( ε )1 + ε Z E J n • ( c ε ( x )) (cid:18) f ( c ε ( x )) + κε + κ √ εg ( ε ) (cid:19) E ε ( x ) d H d − ( x ) ≥ (1 − δ ) I ( E, f ) . If I ( E, f ) = ∞ , for any t >
0, there exists ε such that g ( ε )1 + ε Z E J n • ( c ε ( x )) (cid:18) f ( c ε ( x )) + κε + κ √ εg ( ε ) (cid:19) E ε ( x ) d H d − ( x ) ≥ t . The result follows by choosing the covering associated to this ε and its associated neighborhood as definedin (5.10). We recall that we endow M ( R d ) with the weak topology and B ( R d ) with the topology associatedwith the distance d . We denote by U the basis of neighborhood of the origin of B ( R d ) × M ( R d ) for theassociated product topology. Proposition 6.1 (Lower semi-continuity of the rate function) . The function ( E, ν ) e I ( E, ν ) is lowersemi-continuous, i.e., for any ( E , ν ) ∈ B ( R d ) × M ( R d ) , for any t ≥ such that t < e I ( E , ν ) , thereexists a neighborhood U ( E , ν ) such that for any ( E, ν ) ∈ U ( E , ν ) , we have e I ( E, ν ) ≥ t .Proof. Let ( E , ν ) ∈ B ( R d ) × M ( R d ). Let t > e I ( E , ν ) > t . We claim that there exists aneighborhood U of ( E , ν ) such thatlim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≤ − t . (6.1)We first admit this claim and show how it implies proposition 6.1. Let U be a neighborhood of ( E , ν )such that (6.1) holds. Let ( E, ν ) ∈ U . If e I ( E, ν ) = + ∞ , then we have trivially e I ( E, ν ) ≥ t . If e I ( E, ν ) < ∞ , since U is also a neighborhood of ( E, ν ), then we have using proposition 4.1 − e I ( E, ν ) ≤ lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≤ − t .
56t follows that e I ( E, ν ) ≥ t . Hence, ∀ ( E, ν ) ∈ U e I ( E, ν ) ≥ t . The result follows. It remains to prove the existence of U such that inequality (6.1) holds. We firstconsider the case where ( E , ν ) ∈ T M . By a straightforward application of the proposition 5.1, thereexists a neighborhood U of ( E , ν ) such thatlim sup n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U (cid:1) ≤ − t . It follows that lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≤ − t . We now consider the case where ( E , ν ) / ∈ T M . By proposition 3.1, there exists a neighborhood U of( E , ν ) such that lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ) ≤ − t . To prove theorem 1.2, it is sufficient to prove that e I is a good rate function, a tightness result andthat the local estimates are satisfied (see section 6.2 in [1], even if it is not a real large deviation principle,we can follow exactly the same steps as in section 6.2). Theorem 1.2 is thus a direct consequence of thefollowing proposition. Proposition 6.2.
The function e I is a good rate function. There exist positive constants c and λ suchthat ∀ λ ≥ λ ∀ U ∈ U lim ε → lim sup n →∞ n d − log P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , λ ]) + U (cid:17) ≤ − cλ . Moreover, the following local estimates are satisfied ∀ ( E, ν ) ∈ B ( R d ) × M ( R d ) ∀ U ∈ U lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ ( E, ν ) + U ) ≥ − e I ( E, ν ) , ∀ ( E, ν ) ∈ B ( R d ) × M ( R d ) such that e I ( E, ν ) < ∞ , ∀ ε > ∃ U ∈ U lim ε → lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ ( E, ν ) + U ) ≤ − (1 − ε ) e I ( E, ν ) . Proof of proposition 6.2.
Step 1. We prove that e I is a good rate function. Let us prove that itslevel sets are compact. Set M = (cid:8) ν ∈ M ( R d ) : ν ( V (Ω , ≤ d M H d − (Γ ) , ν ( V (Ω , c ) = 0 (cid:9) . (6.2)The set M is relatively compact for the weak topology by Prohorov theorem. Let us check that this setis closed. Let ( ν n ) n ≥ be a sequence of elements in M that converges weakly towards ν . By Portmanteautheorem, we have ν ( V (Ω , c ) ≤ lim inf n →∞ ν n ( V (Ω , c ) = 0and ν ( R d ) ≤ lim inf n →∞ ν n ( R d ) ≤ d M H d − (Γ ) . It follows that ν ( V (Ω , ≤ d M H d − (Γ )57nd ν ∈ M . The set M is compact for the weak topology. Let λ >
0. By lemma 3.2, there exists β > λ and Ω such that ∀ n ≥ P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) ≤ exp( − λn d − )and solim inf n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − (cid:1) ≤ − λ . (6.3)Let us assume there exists ( E, ν ) ∈ T M such that P ( E, Ω) > β and e I ( E, ν ) ≤ λ . Since F
7→ P ( F, Ω) islower semi-continuous, there exists a neighborhood U of E such that for any F ∈ U , P ( F, Ω) ≥ β . Let U be a neighborhood of ν . By proposition 4.1, we have − e I ( E, ν ) ≤ lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U × U ) . Using that e I ( E, ν ) ≤ λ , it follows that − λ ≤ lim ε → lim inf n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U × U ) ≤ lim inf n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , R( E n ) ∈ U ) ≤ lim inf n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , P (R( E n ) , Ω) ≥ β )= lim inf n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − ) , this contradicts inequality (6.3). Consequently, if e I ( E, ν ) ≤ λ , then P ( E, Ω) ≤ β . It follows that the set e I − ([0 , λ ]) ⊂ C β × M . Since the set C β is compact for the topology associated to the distance d and M is compact for the weak topology, then the set C β × M is compact for the associated product topology.Besides, since e I is lower semi-continuous, its level sets are closed. It follows that e I − ([0 , λ ]) is compact.This implies that e I is a good rate function. Step 2. We prove the e I -tightness. Let λ ≥
0. Let U ∈ U . Let K >
0. Let ε >
0. For any(
E, ν ) ∈ T M such that e I ( E, ν ) < + ∞ , by proposition 5.1, there exists a neighborhood U ( E,ν ) such thatlim sup n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U ( E,ν ) ) ≤ − (1 − ε ) e I ( E, ν ) . For (
E, ν ) such that e I ( E, ν ) ≤ λ , up to taking U ( E,ν ) ∩ ( e I − ([0 , λ ] + U ), we can assume that U ( E,ν ) ⊂ ( e I − ([0 , λ ] + U ). By proposition 5.1, for ( E, ν ) ∈ T M such that e I ( E, ν ) = + ∞ , there exists a neighbor-hood U ( E,ν ) such thatlim sup n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U ( E,ν ) ) ≤ − K .
Using lemma 3.2, there exists β > λ and Ω such thatlim sup n →∞ n d − log P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − (cid:1) ≤ − λ . For (
E, ν ) ∈ C β × M \ T M , by proposition 3.1, there exists a neighborhood U ( E,ν ) such thatlim ε → lim sup n →∞ n d − log P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U ( E,ν ) ) ≤ − K .
The set C β = { E ∈ B ( R d ) : P ( E, Ω) ≤ β } is compact for the topology associated to the distance d . The set M defined in (6.2) is compact for the weak topology. Therefore, we can extract from58 U ( E,ν ) , ( E, ν ) ∈ C β × M ) a finite covering ( U ( E i ,ν i ) ) i =1 ,...,N of C β × M . Let ε >
0. We have P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , λ ]) + U (cid:17) ≤ N X i =1 P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , λ ]) + U, (R( E n ) , µ n ( E n )) ∈ U ( E i ,ν i ) (cid:17) + P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ d M H d − (Γ ) n d − , card( E n ) ≥ βn d − (cid:1) . If e I ( E i , ν i ) ≤ λ , since by construction U ( E i ,ν i ) ⊂ ( e I − ([0 , λ ]) + U ), then P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ ( e I − ([0 , λ ]) + U ) c ∩ U ( E i ,ν i ) (cid:17) = 0 . By lemma 1.11, it follows thatlim ε → lim sup n →∞ n d − log P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , λ ]) + U (cid:17) ≤ − min (cid:16) (1 − ε ) min ne I ( E i , ν i ) : i = 1 , . . . , N, e I ( E i , ν i ) ∈ ] λ, ∞ [ o , K (cid:17) ≤ − min((1 − ε ) λ, K ) . By letting first K go to infinity and then ε go to 0, we obtainlim ε → lim sup n →∞ n d − log P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , λ ]) + U (cid:17) ≤ − λ . Step 3. We prove the local estimates.
Finally, note that the local estimates are direct consequencesof propositions 4.1 and 5.1. This concludes the proof.We now study the lower large deviations for the maximal flow φ n . We recall the definition of J : ∀ λ ≥ J ( λ ) = inf ne I ( E, ν ) : (
E, ν ) ∈ B ( R d ) × M ( R d ) , ν ( R d ) = λ o . Let λ min be defined by λ min = inf { λ ≥ J ( λ ) < ∞} . We first prove this intermediate result on J . Proposition 6.3.
The function J is lower semi-continuous. The function J is finite on ] λ min , φ Ω ] Moreover, J is decreasing on ] λ min , φ Ω ] , J ( λ ) = ∞ for λ ∈ [0 , λ min [ ∪ ] φ Ω , + ∞ [ and ∀ λ ∈ ]0 , φ Ω [ − J ( λ − ) ≤ lim inf n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ lim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ − J ( λ ) , where J ( λ − ) is the left hand limit of J at λ .Proof of proposition 6.3. Step 1. We prove that the infimum in the definition of J is attainedand that the function J has the desired properties. Let λ ∈ [0 , φ Ω ] such that J ( λ ) < ∞ . Let t > J ( λ ). Let β be such that for any ( E, ν ) ∈ T M such that e I ( E, ν ) ≤ t then P ( E, Ω) ≤ β (see proof ofproposition 6.2, step 1). It follows that J ( λ ) = inf ne I ( E, ν ) : (
E, ν ) ∈ C β × M , ν ( R d ) = λ o . Let ( ν n ) n ≥ be a sequence such that for any n ≥ ν n ( R d ) = λ and the sequence weakly convergestowards ν . It yields that ν ( R d ) = λ . Consequently, the set (cid:8) ( E, ν ) ∈ C β × M , ν ( R d ) = λ (cid:9)
59s compact. Since the function e I is lower semi-continuous, it attains its minimum over this set: thereexists ( E, ν ) ∈ T M such that ν ( R d ) = λ and J ( λ ) = e I ( E, ν ).Let λ ∈ ] λ min , φ Ω [. Let ( E, µ ) ∈ T M such that µ ( R d ) = λ ∈ ] λ min , λ [ . We write µ = f H d − | E with( E, f ) ∈ T . Let f E be the function defined by ∀ x ∈ E f E ( x ) = (cid:26) ν G ( n E ( x )) if x ∈ ∂ ∗ E ∩ Ω ν G ( n Ω ( x )) if x ∈ ∂ ∗ Ω ∩ ((Γ \ ∂ ∗ E ) ∪ (Γ ∩ ∂ ∗ E )) (6.4)We set µ E = f E H d − | E . By construction, we have µ E ( R d ) = I Ω ( E ) ≥ φ Ω . Since λ ∈ ] λ , φ Ω [, there exists α ∈ ]0 ,
1[ such that ( αµ + (1 − α ) µ E )( R d ) = λ . By convexity of J v we have J ( λ ) ≤ I ( E, αf + (1 − α ) f E ) ≤ α I ( E, f ) + (1 − α ) I ( E, f E ) ≤ α I ( E, f )where we use the fact that J n E ( x ) ( ν G ( n E ( x )) = 0 and J n Ω ( x ) ( ν G ( n Ω ( x )) = 0. It follows that e I ( E, µ ) ≥ α J ( λ ) > J ( λ ) . (6.5)It follows that for λ ≤ φ Ω , J ( λ ) = inf ne I ( E, ν ) : (
E, ν ) ∈ B ( R d ) × M ( R d ) , ν ( R d ) ≤ λ o . Taking the infimum in (6.5) for any (
E, ν ) such that µ ( R d ) = λ , we obtain J ( λ ) > J ( λ ). Hence, thefunction J is decreasing on ] λ min , φ Ω [. It follows that J is finite on ] λ min , φ Ω ]. We claim that J ( φ Ω ) = 0.By theorem 1.12, the set Σ a is not empty, there exists F ⊂ Ω such that I Ω ( F ) = φ Ω = Cap( F, ν G ( n • ) F ).It is easy to check that ( F, ν G ( n • ) F ) ∈ T and I ( F, ν G ( n • ) F ) = 0 using that J n • ( ν G ( n • )) = 0 (seetheorem 1.13). It follows that J ( φ Ω ) = 0.Moreover for any ( E, ν ) ∈ T M , the condition of minimality implies that ν ( R d ) ≤ Cap(
F, ν G ( n • ) F ) = φ Ω . It follows that for any λ > φ Ω , we have J ( λ ) = + ∞ . Step 2. We prove a lower bound.
Let λ >
0. We aim to prove that ∀ δ > − J ( λ ) ≤ lim inf n →∞ n d − log P (cid:0) φ n (Γ , Γ , Ω) ≤ ( λ + δ ) n d − (cid:1) . (6.6)The result is clear if J ( λ ) = + ∞ . Let us now assume that J ( λ ) < ∞ . Let ( E, ν ) ∈ T M such that e I ( E, ν ) = J ( λ ) and ν ( R d ) = λ . Let g ∈ C c ( R d , R ) such that g = 1 on V (Ω ,
1) and g ≥
0. Hence, wehave ν ( g ) = λ . Let δ >
0. Let U be a neighborhood of E and U = { µ ∈ M ( R d ) : | µ ( g ) − ν ( g ) | ≤ δ } .We have ∀ ε > P ( ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) ∈ U × U ) ≤ P (cid:0) ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ ( λ + δ ) n d − (cid:1) = P ( φ n (Γ , Γ , Ω) ≤ ( λ + δ ) n d − ) . Using proposition 4.1, it follows that − J ( λ ) = − e I ( E, ν ) ≤ lim inf n →∞ n d − log P (cid:0) φ n (Γ , Γ , Ω) ≤ ( λ + δ ) n d − (cid:1) . Step 3. We prove an upper bound.
Let λ ∈ [0 , φ Ω ]. We aim to prove that ∀ δ > n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ − J (min( λ + δ, φ Ω )) . (6.7)Let K > δ > ε >
0. Thanks to proposition 5.1, to each (
E, ν ) ∈ T M such that e I ( E, ν ) < ∞ ,we can associate a neighborhood U ( E,ν ) of ( E, ν ) such thatlim sup n →∞ n d − log P ( ∃E n ∈ C n (Γ , Γ , Ω) : (R( E n ) , µ n ( E n )) ∈ U ( E,ν ) ) ≤ − (1 − ε ) e I ( E, ν ) .
60p to taking U ( E,ν ) ∩ ( B ( R d ) × { µ ∈ M ( R d ) : | µ ( g ) − ν ( g ) | < δ } , we can assume that U ( E,ν ) ⊂ ( B ( R d ) × { µ ∈ M ( R d ) : | µ ( g ) − ν ( g ) | < δ } . We recall that g was defined in the previous step. Since e I is a good rate function, the set e I − ([0 , K ])is compact we can extract from ( U ( E,ν ) , ( E, ν ) ∈ e I − ([0 , K ])) a finite covering ( U ( E i ,ν i ) , i = 1 , . . . , N ) of e I − ([0 , K ]). From lemma 6.6 in [1], there exists a neighborhood U of 0 such that e I − ([0 , K ]) + U ⊂ N [ i =1 U ( E i ,ν i ) . Let ε >
0. It is easy to check that the (Γ n , Γ n )-cutset E minn that achieves the minimal capacity (if thereare several cutsets that achieve the minimal capacity, we choose one according to a deterministic rule)is in C n ( ε ). Using the previous inclusion, we have P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ P ( ∃E n ∈ C n ( ε ) : V ( E n ) ≤ λn d − ) ≤ N X i =1 P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ λn d − , (R( E n ) , µ n ( E n )) ∈ U ( E i ,ν i ) )+ P ( ∃E n ∈ C n ( ε ); (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , K ]) + U ) . Note that on the event {∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ λn d − } , if for i ∈ { , . . . , N } , (R( E n ) , µ n ( E n )) ∈ U ( E i ,ν i ) then we have ν i ( R d ) = ν i ( g ) ≤ µ n ( E n )( g ) + δ ≤ λ + δ . Consequently, for i such that ν i ( R d ) > λ + δ , we have P ( ∃E n ∈ C n (Γ , Γ , Ω) : V ( E n ) ≤ λn d − , (R( E n ) , µ n ( E n )) ∈ U ( E i ,ν i ) ) = 0 . Using proposition 6.2, we havelim ε → lim sup n →∞ n d − log P (cid:16) ∃E n ∈ C n ( ε ) : (R( E n ) , µ n ( E n )) / ∈ e I − ([0 , K ]) + U (cid:17) ≤ − cK . Using lemma 1.11, it follows thatlim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ − min (cid:16) (1 − ε ) min { e I ( E i , ν i ) : ν i ( R d ) ≤ λ + δ, i = 1 , . . . , N } , cK (cid:17) ≤ − min((1 − ε ) J (min( λ + δ, φ Ω )) , cK )where we use that for any i ∈ { , . . . , N } , since ( E i , ν i ) ∈ T M , by the minimality condition, we have ν i ( R d ) ≤ φ Ω . If J (min( λ + δ, φ Ω )) = + ∞ , then by letting K go to infinity, we obtainlim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ −∞ = − J (min( λ + δ, φ Ω )) . If J (min( λ + δ, φ Ω )) < ∞ , then by letting K go to infinity and then ε go to 0, we obtainlim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ − J (min( λ + δ, φ Ω )) . The result follows.
Step 4. We prove that the function J is lower semi-continuous. Let λ >
0. Let ( λ n ) n ≥ bea sequence of non negative real number that converges towards λ . If lim inf n →∞ J ( λ n ) = + ∞ there isnothing to prove. Let us assume that lim inf n →∞ J ( λ n ) < ∞ . Let ψ be an extraction such thatlim n →∞ J ( λ ψ ( n ) ) = lim inf n →∞ J ( λ n ) (6.8)61nd for all n ≥ J ( λ ψ ( n ) ) < ∞ . There exists ( E n , ν n ) ∈ T M such that ν n ( R d ) = λ ψ ( n ) and J ( λ ψ ( n ) ) = e I ( E n , ν n ) . Since the sequence ( J ( λ ψ ( n ) )) n ≥ converges, there exists m > ∀ n ≥ J ( λ ψ ( n ) ) ≤ m . Since e I is a good rate function, the set e I − ([0 , m ]) is compact. We can extract from the sequence( E n , ν n ) n ≥ a sequence ( E φ ( n ) , ν φ ( n ) ) n ≥ that converges towards ( E, ν ) ∈ e I − ([0 , m ]) ( E n converges forthe distance d and ν n converges weakly towards ν ). Since e I is lower semi-continuous, we havelim n →∞ J ( λ ψ ( n ) ) = lim inf n →∞ J ( λ ψ ( φ ( n )) ) = lim inf n →∞ e I ( E φ ( n ) , ν φ ( n ) ) ≥ e I ( E, ν ) . Since ν φ ( n ) weakly converges towards ν , we have ν ( R d ) = lim n →∞ ν φ ( n ) ( R d ) = lim n →∞ λ ψ ( φ ( n )) = λ . Combining the two previous inequalities we obtainlim n →∞ J ( λ ψ ( n ) ) ≥ e I ( E, ν ) ≥ J ( λ ) . By (6.8), we have J ( λ ) ≤ lim inf n →∞ J ( λ n ) . It follows that J is lower semi-continuous on R + . Step 5. Conclusion.
Using inequality (6.6), we have ∀ λ > n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≥ − lim δ → J ( λ − δ ) := − J ( λ − )and using inequality (6.7) and the fact that J is lower semi-continuous ∀ λ ∈ [0 , φ Ω ] lim sup n →∞ n d − log P ( φ n (Γ , Γ , Ω) ≤ λn d − ) ≤ − lim δ → J (min( λ + δ, φ Ω )) ≤ − J ( λ ) . Proof of theorem 1.4.
Note that for any λ ≥ φ Ω , we have J ( λ ) = + ∞ since by definition there does notexist any ( E, ν ) ∈ T M such that ν ( R d ) = λ > φ Ω because of the condition of minimality. • Lower bound.
We prove the local lower bound: ∀ λ ≥ ∀ ε > n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ ] λ − ε, λ + ε [ (cid:19) ≥ − J ( λ ) . Let λ > ε >
0. If J ( λ ) = + ∞ , there is nothing to prove. If λ = φ Ω , since J ( φ Ω ) = 0 and by thelaw of large numbers for φ n (Γ , Γ , Ω) (see theorem 1.12), the result follows. Otherwise, λ < φ Ω and wehave P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ ] λ − ε, λ + ε [ (cid:19) ≥ P (cid:18) φ n (Γ , Γ , Ω) n d − ≤ λ + δ (cid:19) − P (cid:18) φ n (Γ , Γ , Ω) n d − ≤ λ − ε (cid:19) where δ > λ + δ < φ Ω and δ ≤ ε . Since J ( λ + δ ) < J ( λ − ε ), byproposition 6.3 and lemma 1.11, it leads tolim inf n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ ] λ − ε, λ + ε [ (cid:19) ≥ − J (( λ + δ ) − ) ≥ − J ( λ )where we use that J is decreasing on ] λ min , φ Ω [. 62 Upper bound.
We have to prove that for all closed subset F of R + lim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) ≤ − inf F J .
Let F be a closed subset of R + . If φ Ω ∈ F , then inf F J = 0 and the result is obvious. We suppose nowthat φ Ω / ∈ F . We consider F = F ∩ [0 , φ Ω ] and F = F ∩ [ φ Ω , + ∞ [. We claim thatlim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) = lim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) . (6.9)Cerf and Théret proved in theorem 1 in [3] that the upper large deviations of φ n (Γ , Γ , Ω) are of volumeorder. Hence, if F is not empty, we have f = inf F > φ Ω andlim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) = −∞ . This equality trivially holds when F is empty. By lemma 1.11, we deduce equality (6.9). Let us assumethat F is not empty. Let f = sup F . Since F is closed, we have f ∈ F and f < φ Ω , and usingequality (6.9), we havelim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) ≤ lim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ≤ f (cid:19) . Using proposition 6.3, it yields thatlim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) ≤ lim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ≤ f (cid:19) = − J ( f ) = − inf F J since J is decreasing on ] λ min , φ Ω ]. Let us assume that F = ∅ . Using equality (6.9), we havelim sup n →∞ n d − log P (cid:18) φ n (Γ , Γ , Ω) n d − ∈ F (cid:19) = −∞ = − inf F J = − inf F J . • Property of λ min . We claim that for all λ < λ min , there exists N ≥ ∀ n ≥ N P (cid:18) φ n (Γ , Γ , Ω) n d − ≤ λ (cid:19) = 0 . If λ min = 0, we have nothing to prove. Let us assume λ min >
0. We recall that δ G = inf { t : P ( t ( e ) ≤ t ) > } . Note that the function I : F Z F k n • ( x ) k d H d − ( x )is lower semi-continuous. This can be deduced from the lower semi-continuity of the surface energy (seesection 14.2 in [1]). Therefore, the infimum of I is achieved on the following compact set { F ∈ B ( R d ) : F ⊂ Ω , P ( F, Ω) ≤ d H d − (Γ ) } . We denote by F a set that achieves the infimum. Let λ ∈ [ λ min , φ Ω ]. Let ( E, ν ) ∈ T M such that ν ( R d ) = λ and e I ( E, ν ) = J ( λ ). Write ν = f H d − | E . We have f ≥ δ G k n • k H d − -almost everywhere on E , if not, by theorem 1.13, it contradicts the fact that R E J n • ( x ) ( f ( x )) d H d − ( x ) < ∞ . Hence, we have λ ≥ Z E δ G k n • k d H d − = δ G I ( E ) ≥ δ G I ( F ) . It yields that λ min ≥ δ G I ( F ) . ε >
0, we define f ε as follows ∀ x ∈ F f ε ( x ) = (1 + ε ) δ G k n • ( x ) k . (6.10)As long as G is not a dirac mass (the study of large deviations is trivial in that case), we have ∀ v ∈ S d − ν G ( v ) > δ G k v k . Thus, we can choose ε small enough such that ∀ v ∈ S d − (1 + ε ) δ G k v k ≤ ν G ( v ) . We have I ( F , f ε ) = Z F J n • ( x ) ( f ε ( x )) d H d − ( x ) ≤ H d − ( F ) sup v ∈ S d − J v ((1 + ε ) δ G k v k ) < ∞ We claim that ( F , f ε ) is minimal. Let E ⊂ Ω such that P ( E, Ω) < ∞ . We set ∀ x ∈ E g ( x ) = (cid:26) f ε ( x ) if x ∈ F ∩ E ν G ( n • ( x )) if x ∈ E \ F . Let us prove that Cap(
E, g ) ≥ Cap( F , f ε ). We distinguish two cases. We assume first that P ( E, Ω) ≤ d H d − (Γ ). Since F achieves the infimum on F on the set { F ∈ B ( R d ) : F ⊂ Ω , P ( F, Ω) ≤ d H d − (Γ ) } , then I ( F ) ≤ I ( E ) and Z E \ F k n • ( x ) k d H d − ( x ) ≥ Z F \ E k n • ( x ) k d H d − ( x ) . (6.11)Thanks to the choice of ε , we have Z E \ F ν G ( n • ( x )) d H d − ( x ) ≥ (1 + ε ) δ G Z E \ F k n • ( x ) k d H d − ( x ) . (6.12)Combining inequalities (6.11) and (6.12), it yields thatCap( E, g ) ≥ Z E ∩ F f ε ( x ) d H d − ( x ) + Z E \ F ν G ( n • ( x )) d H d − ( x ) ≥ Z E ∩ F f ε ( x ) d H d − ( x ) + Z F \ E (1 + ε ) δ G k n • ( x ) k d H d − ( x ) = Cap( F , f ε ) . Let us assume that P ( E, Ω) > d H d − (Γ ). It follows thatCap( E, g ) ≥ Z E δ G k n • ( x ) k d H d − ( x ) ≥ δ G H d − ( E ) ≥ δ G P ( E, Ω) ≥ dδ G H d − (Γ ) . Besides, we have I ( F ) ≤ I ( ∅ ) = Z Γ k n Ω ( x ) k d H d − ( x ) ≤ d H d − (Γ ) . Hence, we have Cap( F , f ε ) = (1 + ε ) δ G I ( F ) ≤ (1 + ε ) δ G d H d − (Γ ) ≤ Cap(
E, g ) . Finally, ( F , f ε ) is minimal and ( F , f ε ) ∈ T . It follows that λ min ≤ (1 + ε ) δ G I ( F ) . By letting ε go to 0, we obtain that λ min = δ G I ( F ) . λ < λ min . Let us assume there exists an increasing sequence ( a n ) n ≥ such that P (cid:18) φ a n (Γ , Γ , Ω) a d − n ≤ λ (cid:19) > . On the event { φ a n (Γ , Γ , Ω) ≤ λa d − n } we pick E a n ∈ C a n (Γ , Γ , Ω) such that V ( E a n ) ≤ λa d − n . Itfollows that card( E a n ) ≤ λa d − n /δ G . Besides, we have I (R( E a n )) a d − n ≤ card( E a n ) . It follows that R( E a n ) belongs to the compact set { F ⊂ Ω : I ( F ) ≤ λ/δ G } , up to extracting a subse-quence again, we can assume that lim n →∞ d (R( E a n ) , E ) = 0 for some E ⊂ Ω. Since the function I islower semi continuous, it follows that I ( E ) ≤ lim inf n →∞ I (R( E a n )) ≤ λ/δ G < I ( F ) . This contradicts the minimality of F . The result follows. References [1] Raphaël Cerf. The Wulff crystal in Ising and percolation models. In
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