Zero-temperature 2D Ising model and anisotropic curve-shortening flow
aa r X i v : . [ m a t h . P R ] J u l ZERO-TEMPERATURE 2D STOCHASTIC ISING MODEL ANDANISOTROPIC CURVE-SHORTENING FLOW
HUBERT LACOIN, FRANC¸ OIS SIMENHAUS, AND FABIO LUCIO TONINELLI
Abstract.
Let D be a simply connected, smooth enough domain of R . For L > Z with initial condition such that σ x = − x ∈ L D and σ x = +1 otherwise. It is conjectured[23] that, in the diffusive limit where space is rescaled by L , time by L and L → ∞ , theboundary of the droplet of “ − ” spins follows a deterministic anisotropic curve-shortening flow,where the normal velocity at a point of its boundary is given by the local curvature times anexplicit function of the local slope. The behavior should be similar at finite temperature T < T c ,with a different temperature-dependent anisotropy function.We prove this conjecture (at zero temperature) when D is convex. Existence and regularityof the solution of the deterministic curve-shortening flow is not obvious a priori and is part ofour result. To our knowledge, this is the first proof of mean curvature-type droplet shrinkingfor a model with genuine microscopic dynamics.2000 Mathematics Subject Classification: 60K35, 82C20Keywords: Ising model, Glauber dynamics, Curve-shortening flow.
Contents
1. Introduction 12. Model and results 33. Local interface dynamics 84. Proof of Theorem 2.3: evolution of the scale-invariant droplet 145. Proof of Theorem 2.1: existence of anisotropic curve-shortening flow with convexinitial condition 266. Proof of Theorem 2.2: evolution of a convex droplet 337. Proof of Theorem 3.2: scaling limit for SSEP 39Appendix A. Proof of Theorem 3.4: scaling limit for the zero-range process 41Acknowledgments 52References 521.
Introduction
Consider a thermodynamic system with two coexisting phases and imagine to prepare itin an initial condition where a droplet of one phase is immersed in the other phase. If thesystem undergoes a dynamics that does not conserve the order parameter, it is well understoodphenomenologically [20] that the droplet will shrink in order to decrease its surface tensionuntil it eventually disappears, and that (roughly speaking) the normal speed at a point of its
F. T. was partially supported by European Research Council through the “Advanced Grant” PTRELSS 228032and by ANR project SHEPI.
H. LACOIN, F. SIMENHAUS, AND F. L. TONINELLI boundary will be proportional to the local mean curvature. Deriving such behavior from firstprinciples, i.e. from a microscopic model undergoing a local (stochastic) dynamics, is a muchharder task and this program was started only rather recently [23]. More precisely, what oneexpects is that, if the initial droplet is of diameter L , it will “disappear” within a time of order L (this behavior is sometimes referred to as “Lifshitz law”). Moreover, in the “diffusive limit”where L → ∞ and at the same time space is rescaled by L (so that the initial droplet is ofsize O (1)) and time is accelerated by L , the droplet evolution should become deterministic andfollow some anisotropic version of a mean curvature flow. Anisotropy (i.e. the fact that thenormal velocity will also depend on the local orientation of the droplet boundary) is expectedwhen the underlying model is defined on a lattice, as will be the case for us.Up to now, mathematical progress on this issue has been rather modest, the main difficultybeing that it is not clear how to implement the idea that the fast modes related to relaxationinside the two pure phases should decouple from slow modes related to the interface motion,which are responsible for the diffusive time scaling L .A fairly well understood situation is that where the interface can be described by a heightfunction and the bulk structure of the two phases is disregarded. This is possible (by definition)for the so-called “effective interface models” or Ginzburg-Landau ∇ φ interface models: for mod-els with continuous heights and strictly convex potential undergoing a Langevin-type dynamics,Funaki and Spohn [10] derived the full mean-curvature motion in the diffusive scaling. Anotherwell-studied case is that of models with Kac-type potentials: in this case, mean-curvature mo-tion can be proven to emerge [5, 6, 15] in a limit where interaction range is taken to infinity atsome stage, but in this limit there is no sharp interface separating the phases and the systembecomes very close to mean-field.As for true lattice models, results are much more scarce. For instance, for the two-dimensionalnearest-neighbor Ising model below the critical temperature, the best known upper bound onthe “disappearance time” for a droplet of “ − phase” immersed in the “+ phase” is of the order L c ( T ) log L [22], very far from the expected L scaling. Recently, a weak version of the Lifshitzlaw was proven for the three-dimensional Ising model at zero temperature: the disappearancetime of a “ − ” droplet is of order L (upper and lower bounds), up to multiplicative logarithmic(in L ) corrections [2]. When the dimension is higher than three (always at zero temperature),an upper bound for the disappearance time of order L (log L ) c , for some constant c , was provenin [18].In this work, we concentrate on the two-dimensional nearest-neighbor Ising model on theinfinite square lattice. The dynamics takes a very simple form: each spin is updated with rateone and after the update it takes the same value as the majority of its neighbors, or the value ± − ”. In this case,the disappearance time of a large “ − ” droplet should be asymptotically given by one half itsvolume (number of “ − ” spins). Moreover, in the diffusive scaling limit the droplet boundaryshould be given by a deterministic curve γ ( t ) whose normal speed is given by the local (signed)curvature, times a function a ( θ ) where θ is the angle of the local normal vector. The function a ( · ) is explicitly known, see (2.4). In this two-dimensional setting, it is more natural to refer tosuch flow as “(anisotropic) curve-shortening flow” (rather than “mean curvature flow”).Our main result (Theorem 2.2) is a proof of the curve-shortening conjecture (and, as a byprod-uct, of the Lifshitz law) when the initial droplet is convex.There are some previous partial results available on this problem. The scaling limit of theevolution when initially spins are “ − ” in the first quadrant of Z (infinite corner) and “+”elsewhere is described in [16, Section 4.2] (with the language of exclusion processes rather thanspin systems). This is a simple situation because the interface motion is mapped to symmetric URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 3 simple exclusion and is described by the associated height function at all times. In [23], Spohndescribed the scaling limit of the interface motion in a situation that more or less corresponds tothe zero-temperature Ising model in an infinite vertical cylinder, with an initial condition suchthat the interface separating “+” from “ − ” spins can be globally described by a height functionat all times (in particular, this cannot describe a droplet, and implicitly he has to modify thedynamics to guarantee that droplets do not appear later in the evolution). In [4], Chayes et al. proved the Lifshitz law (but not the curve-shortening conjecture) for a modified dynamics whereupdates which break the droplet into several droplets are forbidden. In [3], Cerf and Louhichicomputed the “drift at time 0” of the droplet (for the non-modified dynamics), but their resultdoes not allow to get information on the evolution for finite time t > a ( · ) is not smooth (its derivative has jumps, reflecting the singularities of the surface tensionat zero temperature), while the existing results assume that a ( · ) is at least C , cf. [11, 12]. Toprove existence, uniqueness and regularity of the solutions (cf. Theorem 2.1), we will regularizethe function a ( · ) and then analyze the regularized flow following the ideas of [11, 12]. Of course,it will be crucial to guarantee that all the estimates we need are uniform in the regularizationparameter, which tends to zero in the end.The case where the initial “ − ” droplet is non-convex will be considered in future work. Theadditional difficulties are two-fold. First of all, from the analytic point of view, available globalexistence and regularity results for the solution of curve-shortening flows with non-convex initialcondition seem to be limited to the isotropic case where a ( · ) ≡ Model and results
Glauber dynamics and expected limiting evolution.
Set Z ∗ := Z + := { x +(1 / | x ∈ Z } . We consider the zero-temperature stochastic Ising model on ( Z ∗ ) with itsusual lattice structure ( x and y are linked if | x − y | = 1 for the l norm). This is a continuoustime Markov chain ( σ ( t )) t ≥ on the space of spin configurations on ( Z ∗ ) , Ω := {− , } ( Z ∗ ) .We write σ ( t ) = ( σ x ( t )) x ∈ ( Z ∗ ) and for simplicity we write σ x = − (resp. σ x = +) instead of σ x = − σ x = +1). H. LACOIN, F. SIMENHAUS, AND F. L. TONINELLI
The transition rules are the following : for each site x ∈ ( Z ∗ ) , the value σ x of the spin at x is updated independently with rate one. When the spin at a site is updated, it takes the samevalue as the spin of the majority of its neighbors, or the values ± / − ” spins. That these rules yield a well-definedMarkov chain even in infinite volume is a standard fact (cf. [21]). In what follows (cf. (2.1)), wewill consider only initial conditions where the number of “ − ” spins is finite. It is easy to realizethat the spins outside the smallest square containing all the initial “ − ” spins stay “+” forever,so that in reality we have a dynamics on a finite volume and the question of existence of theprocess is trivial.We are interested in the evolution of the set of ” − ” spins for this Markov chain when theinitial condition σ (0) is a large droplet, i.e. a finite connected set of ” − ” spins surrounded by”+” spins. In that case, almost surely, after a finite time τ + , all the ” − ” spins have turnedto ”+” and the dynamics will stay forever in the all ”+” configuration (which is an absorbingstate). Our aim is to describe the evolution of the shape of the rescaled “ − ” droplet on a proper(diffusive) time-scale. In the next paragraph we make that aim more precise.We consider a compact, simply connected subset D ⊂ [ − , whose boundary is a closedsmooth curve. Given L ∈ N we consider the Markov chain described above with initial condition σ x (0) = ( − x ∈ ( Z ∗ ) ∩ L D , +1 otherwise . (2.1)In order to see a set of ” − ” spins as a subset of R , each vertex x ∈ ( Z ∗ ) may be identifiedwith the closed square of side-length one centered at x , C x := x + [ − / , / . (2.2)One defines A L ( t ) := [ { x : σ x ( t )= − } C x , (2.3)which is the “ − droplet” at time t for the dynamics. The boundary of A L ( t ) is a union of edgesof Z (this is the only reason why we defined the Ising model on ( Z ∗ ) ).What was conjectured by Lifshitz [20] on heuristic grounds for the low temperature Isingmodel is that A L ( t ) should follow an anisotropic curve shortening motion: after rescaling spaceby L and time by L and letting L tend to infinity, the motion of the interface between A L ( t ) andits complement should be deterministic and the local drift of the interface should be proportionalto the curvature, with an anisotropic correction to reflect anisotropy of the underlying lattice.More precisely, one can formulate this conjecture as follows [23]: Let γ ( t, L ) denote the boundaryof the (random) set (1 /L ) A L ( L t ). Then, for L → ∞ , γ ( t, L ) should converge to a deterministiccurve γ ( t ) and the evolution of ( γ ( t )) t ≥ should be such that the normal velocity at a point x ∈ γ ( t ) is given by the curvature at x , times an anisotropic factor a ( θ x ), where θ x is the slopeof the outwards directed normal to γ ( t ) at x . The velocity is directed inwards at points where γ ( t ) is convex and outwards at points where it is concave. The function a ( · ) should have theexplicit expression a ( θ ) := 12( | cos( θ ) | + | sin( θ ) | ) . (2.4)In particular, the curve γ ( t ) should shrink to a point in a finite time t = Area ( D ) R π a ( θ )d θ = Area ( D )2 . URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 5
Note that the function a ( · ) is symmetric around 0 and is periodic with period π/
2, whichreflects the discrete symmetries of the lattice ( Z ∗ ) . It is important to note for the followingthat a ( · ) is C ∞ except at θ = jπ/ , j = 0 , . . . , a ( θ ) ∼ / − | θ − iπ/ | for θ close to iπ/ , i = 0 , . . . , Results.
Convex initial droplet.
We prove the anisotropic curve shortening conjecture in the casewhere the initial droplet is convex and suitably smooth. Given a strictly convex smooth domain D in R and letting γ = ∂ D be its boundary, we parameterize it following a standard conventionof convex geometry (cf. e.g. [12] and Figure 1). For θ ∈ [0 , π ] let v ( θ ) be the unit vector formingan anticlockwise angle θ with the horizontal axis and let h ( θ ) = sup { x · v ( θ ) , x ∈ γ } (2.5)with · the usual scalar product in R .Sometimes, we abusively say that γ is a convex curve if the domain D it encloses is convex,and we identify γ with D . PSfrag replacements θ x ( θ ) k ( θ ) v ( θ ) h ( θ ) Figure 1.
A graphic description of the support function h . Given θ , considerthe point x ( θ ) of γ that maximizes x · v ( θ ) (it is unique if the curve is strictlyconvex). Then h ( θ ) = x ( θ ) · v ( θ ), and k ( θ ) is the norm of the curvature vector of γ (bold vector) at x ( θ ). If the tangent to γ at x exists it is normal to v ( θ ) and | h ( θ ) | is the distance between the tangent and the origin. We emphasize thatthis construction works equally well when the origin is not inside γ .The function θ h ( θ ) (called “the support function”) uniquely determines γ : D = ∩ ≤ θ ≤ π { x ∈ R : x · v ( θ ) ≤ h ( θ ) } . (2.6)With this parameterization, the anisotropic curve shortening evolution reads ( ∂ t h ( θ, t ) = − a ( θ ) k ( θ, t ) h ( θ,
0) = h ( θ ) (2.7)where, for a convex curve γ , k ( θ ) ≥ x ( θ ) ∈ γ where the outwardnormal forms an anticlockwise angle θ with the horizontal axis and the t -derivative is taken at H. LACOIN, F. SIMENHAUS, AND F. L. TONINELLI constant θ (see [12, Lemma 2.1] for a proof of (2.7)). Of course h ( · ) is the support function of ∂ D .In general, even proving the existence of a solution of (2.7) with a ( · ) given in (2.4) is non-trivial, since a ( · ) has points of non-differentiability and the existing literature (e.g. [11, 12])usually assumes that a ( · ) is at least C .Our first result is Theorem 2.1.
Let
D ⊂ [ − , be strictly convex and assume that its boundary γ = ∂ D isa curve whose curvature [0 , π ] ∋ θ k ( θ ) defines a positive, π -periodic, Lipschitz function.Then there exists a unique flow of convex curves ( γ ( t )) t with curvature defined everywhere, suchthat γ (0) = γ and that the corresponding support function h ( θ, t ) solves (2.7) for t ≥ andsatisfies the correct initial condition h ( θ,
0) = h ( θ ) . The curve γ ( t ) shrinks to a point x f ∈ R at time t f = Area ( D ) / . For t < t f , γ ( t ) is a smooth curve in the following sense: its curvaturefunction k ( · , t ) is Lipschitz and bounded away from and infinity on any compact subset of [0 , t f ) . We let D ( t ) denote the convex closed set enclosed by γ ( t ) (of course, D ( t = 0) = D ). Also,we use the convention that D ( t ) = { x f } if t ≥ t f .For δ > B ( x, δ ) denote the ball of radius δ centered at x and for any compact set C ⊂ R define C ( δ ) := [ x ∈C B ( x, δ ) , C ( − δ ) := [ x/ ∈C B ( x, δ ) ! c . (2.8)Note that D ( t ) ( δ ) = B ( x f , δ ) and D ( t ) ( − δ ) = ∅ if t ≥ t f .An event B L is said to occur with high probability (w.h.p.) if lim L →∞ P ( B L ) = 1. Theorem 2.2.
Under the same assumptions on D as in Theorem 2.1, for any δ > one hasw.h.p. D ( − δ ) ( t ) ⊂ L A L ( L t ) ⊂ D ( δ ) ( t ) for every ≤ t ≤ t f + δ (2.9) A L ( L t ) = ∅ for every t > t f + δ. (2.10) In particular, one has the following convergence in probability: lim L →∞ τ + L Area ( D ) = 12 . (2.11)The reason why in Theorems 2.1 and 2.2 we do not content ourselves with, say, initial C ∞ curves is that, as we see in next section, there is a very natural initial condition whose curvaturefunction is only Lipschitz and not C (and stays so at later times).Theorem 2.2 does not apply directly if one considers D = [0 , or any other non-smooth ornon-strictly convex convex set. However, approximating D from above and below by smoothcompact sets and using monotonicity (cf. Section 2.3), one sees easily that (2.11) holds in anycase. In particular, the disappearance time of an L × L square droplet is with high probability L / o (1)).Theorems 2.2 and 2.3 tell us that for our choices of initial configuration, the disappearancetime of the minus droplet is non-random at first order. This implies that the variation distanceof our Markov Chain from equilibrium (which is concentrated on the all-plus configuration)drops abruptly from 1 to 0 around time L t f within a time-window of width o ( L ) ≪ L t f URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 7 (we conjecture that the correct order of the window should be O ( L / )). This is a particularinstance of a phenomenon called cut-off (cf. [7] and [19]).2.2.2. Scale-invariant droplet.
A particular case of Theorem 2.2 is that where the initial con-dition is scale invariant, i.e. when the limiting evolution ( γ ( t )) t is a homothetic contraction.Consider the function f : (cid:20) − √ , + 1 √ (cid:21) ∋ x f ( x ) = β (cid:26) αx Z x e αt dt − e αx (cid:27) , (2.12)where α is the unique positive solution of4 √ αe − α Z / √ e αt dt = 1 (2.13)and β = −√ e − α < . (2.14)Note that f is C ∞ , positive, concave, symmetric around 0 and increasing on [ − √ , e , e ) the canonical basis of R and ( f , f ) = ( e − e √ , e + e √ ) the image of ( e , e ) bythe rotation of angle − π/
4. We also define the curve γ to be the graph of f in the coordinatesystem ( f , f ), i.e. γ := (cid:26) x f + f ( x ) f (cid:12)(cid:12) x ∈ (cid:20) − √ , √ (cid:21)(cid:27) . (2.15)If S (resp. S ) denotes the symmetry with respect to the axis e (resp. e ) one defines theclosed curve γ by γ = γ ∪ ( S γ ) ∪ ( S γ ) ∪ (( S ◦ S ) γ ) . (2.16)In the sequel D denotes the compact, convex set enclosed in γ , see Figure 2.PSfrag replacements − √ √ √ √ xf ( x ) e e f f γ γ D Figure 2.
The curve γ = ∂ D and the coordinate systems ( e , e ) and ( f , f ). H. LACOIN, F. SIMENHAUS, AND F. L. TONINELLI
One can check that the curvature function θ k ( θ ) of ∂ D is Lipschitz and bounded awayfrom zero, but not differentiable at θ = iπ/ , i = 0 , , ,
3. In this case, Theorem 2.2 can beformulated as follows.
Theorem 2.3.
Assume that D = D . For any η > , w.h.p, ( √ − αt − η ) D ⊂ L A L ( tL ) ⊂ ( √ − αt + η ) D for every t ≥ where we work with the convention that √ x = 0 for x ≤ and that x D = ∅ for x < . Moreover,one has the following convergence in probability: lim L →∞ τ + Area ( L D ) = α lim L →∞ τ + L = 12 . (2.18)It is easy to check, using Lemma 3.6 below and a couple of integrations by parts, that Area ( D ) = 1 /α , yielding the first equality in (2.18). The expression (2.12) for the invariantshape appears also, although with different notations, in the recent work [17].2.3. Graphical construction of the dynamics and monotonicity.
Before starting theproofs, we wish to give a construction of the Markov process (called sometimes the graphicalconstruction ) that yields nice monotonicity properties. We consider a family of independentPoisson clock processes ( τ x ) x ∈ ( Z ∗ ) . More precisely, to each site x ∈ ( Z ∗ ) one associates arandom sequence (independently from other sites) of times ( τ xn ) n ≥ , that are such that τ x = 0and ( τ xn +1 − τ xn ) n ≥ are IID exponential variables with mean one. One also defines randomvariables ( U n,x ) n ≥ , x ∈ ( Z ∗ ) that are IID Bernoulli variables of parameter 1 /
2, with values ± ξ ∈ {− , } ( Z ∗ ) one constructs the dynamics σ ξ ( t ) startingfrom σ ξ (0) = ξ as follows • ( σ x ( t )) t ≥ is constant on the intervals of the type [ τ xn , τ xn +1 ). • σ x ( τ xn ) is chosen to be equal to ± x satisfies σ y ( τ xn ) = ±
1, and U n,x otherwise (this definition makes sense as, almost surely, twoneighbors will not update at the same time.)This construction gives a simple way to define simultaneously the dynamics for all initialconditions (we denote by P the associated probability). Moreover this construction preservesthe natural order on {− , +1 } ( Z ∗ ) , given by ξ ≥ ξ ′ ⇔ ξ x ≥ ξ ′ x for every x ∈ ( Z ∗ ) (2.19)(this order is just the opposite of the inclusion order for the set of ” − ” spins, which is thereforealso preserved). Indeed, if ξ ≥ ξ ′ , with the above construction, one has P -a.s. ∀ t > σ ξ ( t ) ≥ σ ξ ′ ( t ) . (2.20)3. Local interface dynamics
One problem one has to deal with when proving mean curvature motion for the whole dropletis that even though initially the interface between ”+” and ” − ” (i.e. the geometric boundary ofthe set A L (0)) is a simple curve, it can later split to form several loops. In fact, as a byproductof our results, we will obtain that, with large probability, only very small extra loops can becreated. We will tackle this problem by introducing some auxiliary dynamics that do not allowcreation of new loops and stochastically compare to the original one.A second problem is that the interface that one has to control is not exactly the graph offunction, for which it would be easier to describe the macroscopic motion using partial differential URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 9 equations. We begin by studying two dynamics for which the interface is indeed a graph, andwhich have locally the same large-scale behavior as the true evolution. It is more natural tointroduce these dynamics as dynamics on interfaces rather than dynamics on spins. Our taskthen will consist of glueing together the “local results” of Theorems 3.2 and 3.4 to get Theorems2.2 and 2.3.3.1.
Local dynamics away from the poles and simple exclusion process.
The firstauxiliary dynamics is used to control the evolution of the boundary of (1 /L ) A L ( tL ) away fromthe points (the poles) where the tangent to the deterministic curve γ ( t ) is either horizontal orvertical. The evolution near the poles will be analyzed via a second auxiliary dynamics, seeSection 3.2.Given two positive natural numbers M, N consider the state-space Ω
M,N of nearest-neighbordirected paths of length L := M + N with M steps up and N steps down:Ω M,N = (cid:8) ( h x ) x ∈{ ,...,M + N } ∈ Z M + N +1 (cid:12)(cid:12) | h x +1 − h x | = 1 , h = 0; h M + N = M − N (cid:9) . (3.1)Given h ∈ Ω M,N and x ∈ { , . . . , L − } , we denote by h ( x ) the path with a corner “flipped”at x defined by h ( x ) y = h y for all y = x and h ( x ) x := h x − h x ± = h x − ,h x + 2 if h x ± = h x + 1 ,h x if | h x +1 − h x − | = 2 . (3.2)The dynamics on Ω M,N we consider is the one that flips every corner with rate 1 /
2. Moreprecisely it is the Markov chain whose generator L is defined as L f ( h ) := 12 L − X x =1 ( f ( h ( x ) ) − f ( h )) , ∀ f : Ω M,N R . (3.3)We denote by ( h ( t )) t ≥ the trajectory of the Markov chain started from initial condition h (0) := h ∈ Ω M,N . Remark 3.1.
Note that this dynamics is in one-to-one correspondence with the Ising dynamicson a rectangle N × M with “+” boundary conditions on two adjacent sides and “ − ” boundaryconditions on the two opposite sides, provided that the initial configuration is such that thelength of the − / + boundary is M + N (i.e. the minimal possible length). More precisely(see Figure 3) the correspondence is obtained by taking the graph of h , rotating it by π/ √ h ∈ Ω M,N with a continuous function F : [0 , M + N ] R such that F ( x ) = h x for x = 0 , , . . . , M + N and F ( · ) is affine on intervals( n, n + 1) with integer n .This corner-flip dynamics has been widely studied (see e.g. [24]) and can be mapped to thesymmetric simple exclusion process (SSEP) on a finite interval (just say that there is a particleat x = 0 , . . . , M + N − h x +1 − h x = +1, and check that dynamics in terms ofparticles coincides with that of SSEP). From hydrodynamic-limit results, it is quite clear thatthe rescaled version of h when M, N tends to infinity should satisfy the heat equation (see [16,Section 4.2.] for an account on hydrodynamic equations for the exclusion process). However,we have not found in the literature a proof of the following precise statement we need (we givea concise proof of it in Section 7):
PSfrag replacements MN M + NM − N Figure 3.
One-to-one correspondence between the dynamics in a rectangle withmixed boundary conditions and the corner-flip dynamics on paths. A possiblespin update together with the equivalent corner-flip are represented
Theorem 3.2.
Given a -Lipschitz function φ : [0 , R with φ (0) = 0 , let ( h ( t )) t ≥ thedynamics starting from initial condition h ∈ Ω M L ,N L given by h x := 2 ⌊ Lφ ( x/L ) / ⌋ for even xh x := 2 ⌊ ( Lφ ( x/L ) − / ⌋ + 1 for odd x (3.4) ( M L and N L are implicitly fixed by L and φ (1) ). For all T ≥ and ε > , w.h.p. sup t ∈ [0 ,T ] ,x ∈ [0 , L (cid:12)(cid:12) h ⌊ xL ⌋ ( L t ) − Lφ ( x, t ) (cid:12)(cid:12) ≤ ε (3.5) where φ : [0 , × R + → R is the solution of the Cauchy problem ∂ t φ ( x, t ) = ∂ x φ ( x, t ) ∀ t > , ∀ x ∈ (0 , ,φ (0 , t ) = 0 , φ (1 , t ) = φ (1) ∀ t > ,φ ( x,
0) = φ ( x ) ∀ x ∈ (0 , . (3.6)Here, ⌊ x ⌋ denotes the integer part of x , and the fact that h does belong to Ω M L ,N L is an easyconsequence of φ being 1-Lipschitz.3.2. Local dynamics around the poles and a zero-range process.
For the definition ofthe second auxiliary dynamics, we use the same notation as in the previous section, but noconfusion should arise as the proofs will be given in two independent sections. The state spaceis Ω L := { h : {− L, . . . , L + 1 } 7→ Z } . (3.7)For h ∈ Ω L and x ∈ {− L + 1 , . . . , L } define h + ,x (resp. h − ,x ) as the configuration such that h + ,xy = h y if y = x and h + ,xx = h x + 1 (resp. h − ,xx = h x − h ( t )) t ≥ started from some h ∈ Ω L and with generator L defined by L f ( h ) = 12 L X x = − L +1 c + ,x ( h )( f ( h + ,x ) − f ( h )) + c − ,x ( h )( f ( h − ,x ) − f ( h )) (3.8)where c + ,x ( h ) = { h x +1 >h x } + { h x − >h x } ,c − ,x ( h ) = { h x +1 Note that the values h − L and h L +1 are fixed in time and should be considered as boundaryconditions. Remark 3.3. This dynamics corresponds to the motion of the interface for a modified Isingdynamics in a vertical strip of width 2 L with the following boundary condition: spins on theleft (resp. right) boundary of the system are ”+” if and only if their vertical coordinate islarger than h − L (resp. h L +1 ). The dynamics is modified in the sense that updates are discardedif after the update the boundary between the ” − ” and ”+” domain is not a simple (open)curve (see Figure 4). It is at times more convenient to identify h ∈ Ω L with a c`adl`ag function H : [ − L − / , L + 3 / Z which equals identically h n on intervals [ n − / , n + 1 / 2) for integer n . PSfrag replacements+ spins − spins Figure 4. An example of spin update that splits the interface into two discon-nected components. The interface dynamics presented in this section does notallow this kind of move.Another way to interpret this dynamics [23] is to look at the gradients η x = h x +1 − h x : onerecognizes then a zero-range process with two type of particles (if η x = n > n particles of type A at x , if η x = − n < n type-B particles). Each particleperforms a symmetric simple random walk with jump rate 1 / (2 n ) (with n the occupation numberof the site where the particle sits) to either left or right and particles of different type annihilateinstantaneously when they are at the same site. See Figure 5. Figure 5. Correspondence between interface dynamics and zero-range process.Arrows represent possible motions for the interface and their representation interms of particle moves. When an A particle jumps on a B particle (green arrow)both annihilate.In [23, Appendix A], this dynamics was considered but in a periodized setup. A scalinglimit result was given but the proof there is somewhat sketchy. Here we adapt the proof to thenon-periodic case and write it in full details. Consider φ : [ − , R a C function with φ (1) = φ ( − 1) = 0. We further assume that φ has a finite number of changes of monotonicity. Define Φ : {− L, . . . , L + 1 } 7→ R asΦ ( x ) := Lφ ( x/L ) (3.10)and h : {− L, . . . , L + 1 } 7→ Z by h x := ⌊ Φ ( x ) ⌋ . (3.11)We define Φ : {− L, . . . , L + 1 } × R + → R as the solution of the following Cauchy problem: ∂ t Φ( x, t ) = [ σ ( q x ( t )) − σ ( q x − ( t ))] , Φ( L + 1 , t ) = Φ( − L, t ) = 0 , Φ( x, 0) = Φ ( x ) (3.12)for every t ≥ x ∈ {− L, . . . , L + 1 } , where σ ( u ) = u/ (1 + | u | ) and q x ( t ) : = Φ( x + 1 , t ) − Φ( x, t ) . (3.13)The result we state now is slightly weaker than Theorem 3.2 as it allows to control the profile h only at a fixed time and not on a whole time interval. Theorem 3.4. Given φ as above, consider ( h ( t )) t ≥ the dynamics described by (3.8) with initialcondition h as in (3.11) . Then for any t , the following convergence holds in probability lim L →∞ max x ∈{− L,...,L +1 } L (cid:12)(cid:12) h x ( L t ) − Φ( x, L t ) (cid:12)(cid:12) = 0 . (3.14)It is quite intuitive that one should have that L Φ( ⌊ Lx ⌋ , L t ) → φ ( x, t ) for any x ∈ [ − , φ : [ − , × R + → R is the solution of ∂ t φ ( x, t ) = ∂ x φ ( x,t )(1+ | ∂ x φ ( x,t ) | ) φ (1 , t ) = φ ( − , t ) = 0 φ ( x, 0) = φ ( x ) (3.15)for t ≥ x ∈ ( − , φ : [ − , × R + → R to be the solution of ∂ t ¯ φ ( x, t ) = ∂ x ¯ φ ( x, t )¯ φ (1 , t ) = ¯ φ ( − , t ) = 0¯ φ ( x, 0) = φ ( x ) . (3.16)Then Corollary 3.5. Let φ be as above, and assume further that it is concave with k ∂ x φ k ∞ ≤ η .For every t ≥ and every ε > the following inequality holds w.h.p. ¯ φ ( x/L, t ) − ε ≤ L h x ( L t ) ≤ ¯ φ ( x/L, (1 + η ) − t ) + ε for every x ∈ {− L, . . . , L + 1 } . (3.17) Proof. The result follows by combining Theorem 3.4, Proposition A.9, and by taking limits ofrescaled versions of Φ and Φ in (A.54) when L tends to infinity (cf. Lemma 7.1). (cid:3) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 13 About the scale-invariant shape. Now that we know how the interface should evolvelocally (from Theorems 3.2 and 3.4) it is possible to explain why D should be scale invariant. Bysymmetries of the problem and the fact that motion is driven by curvature, the scale-invariantshape should be convex symmetric around the axes R e , R e . Therefore it is enough to considerthe boundary of the intersection of D with the first quadrant.From Theorem 3.2, if f is a Lipschitz function and ∂ D is the graph of f in the coordinatesystem ( f , f ), the initial drift in the f direction is (1 / ∂ x f, where the factor 1 / / 2) is due to the fact that in the correspondence between Ising dynamics and dynamics ofnearest-neighboring paths, space has to be rescaled by √ 2, cf. Remark 3.1). One the other hand,the homothetic contraction of a shape D of initial velocity α gives an initial drift of the interfacein the f direction α ( − f + x∂ x f ) . (3.18)That leads to the partial differential equation ∂ x f = 4 α ( − f + x∂ x f ) . (3.19)Next we impose the correct boundary conditions on f : • We fix the scaling by imposing that the point (1 , 0) (and therefore also (0 , , ( − , , (0 , − ∂ D . This gives f (cid:16) ± / √ (cid:17) = 1 / √ . (3.20) • To guarantee that the curvature of ∂ D is well defined at the point (0 , 1) we have toimpose ∂ x f (cid:16) − / √ (cid:17) = − ∂ x f (cid:16) / √ (cid:17) = 1 . (3.21)We finally notice that Lemma 3.6. The function f defined in (2.12) is the unique solution of the Cauchy problem (3.19) - (3.20) - (3.21) for x ∈ ( − / √ , +1 / √ . For other values of α the above problem has nosolution.Proof. Uniqueness of the solution is standard from theory of ordinary differential equation. Therest is just a matter of checking. (cid:3) Organization of the paper. Instead of proving directly Theorem 2.2 and then deducingTheorem 2.3 as a corollary, we decided for pedagogical reasons to give first the proof in thecase of the scale-invariant droplet and then to point out what needs to be modified in the moregeneral case of a convex droplet. The reason is that, this way, we can easily separate the questionof comparing the stochastic evolution with the deterministic one (which works more or less thesame in the two cases but is simpler for the invariant droplet, due to its symmetries) from theanalytic, PDE-type issues which appear only in the general case.The paper is therefore organized as follows: • in Section 4, we show that to prove Theorem 2.3 it is sufficient to have a good controlon the continuity of the interface motion (Proposition 4.2) and a result on the evolutionafter an “infinitesimal time” εL (Proposition 4.1). Such crucial results are proven inSections 4.3 and 4.4; • in Section 5 we first prove Theorem 2.1 on the existence of a solution to (2.7), and thenwe prove Theorem 2.2 via a suitable generalization of Propositions 4.2 and 4.1; • finally, the hydrodynamic limit results of Theorems 3.2 and 3.4 are proven in detail inSection 7 and the Appendix A respectively. Proof of Theorem 2.3: evolution of the scale-invariant droplet Reducing to an “infinitesimal” time interval. We decompose the proof of Theorem2.3 into two propositions. The first (and the main one) says that after a time εL the dropletlooks very much the same but contracted by a factor (1 − αε + o ( ε )). Proposition 4.1. For all δ > there exists ε ( δ ) > such that for all < ε < ε ( δ ) , w.h.p., A L ( L ε ) ⊂ (1 − ε ( α − δ )) L D , (4.1) and A L ( L ε ) ⊃ (1 − ε ( α + δ )) L D . (4.2)The second proposition controls continuity in time of the rescaled motion: Proposition 4.2. For every δ > , w.h.p., A L ( L t ) ⊂ (1 + δ ) L D for every t ≥ . (4.3) Moreover, for every δ > there exists ε > such that w.h.p A L ( L t ) ⊃ (1 − δ ) L D for every t ∈ [0 , ε ] . (4.4) Proof of Theorem 2.3 assuming Propositions 4.1 and 4.2. Given η fix δ small enough and ε <ε ( δ ). Then using (4.1) one gets that w.h.p. A L ( L ε ) ⊂ (1 − ( α − δ ) ε ) L D . (4.5)Let ( A (1) L ( L t )) t ≥ denote the evolution of the set of ” − ” spins for the dynamics started frominitial condition ” − ” on (1 − ε ( α − δ )) L D and ”+” elsewhere. Then using the Markov prop-erty and monotonicity of the dynamics, one can couple the dynamics ( A L ( L ( ε + t ))) t ≥ and( A (1) L ( L t )) t ≥ such that on the event (4.5) A L ( L ( ε + t )) ⊂ A (1) L ( L t ) , for every t ≥ . (4.6)Therefore, after conditioning to the event in (4.5) and using (4.1) for (1 − ( α − δ ) ε ) L instead of L , one gets that w.h.p.: A L ( L ε (1 + (1 − ( α − δ ) ε ) )) ⊂ A (1) L ( L (1 − ( α − δ ) ε ) ε ) ⊂ (1 − ( α − δ ) ε ) L D . (4.7)Here we used the fact that A (1) L ( t ) has the same law as A L (1 − αε ) ( t ). Using this argumentrepeatedly one gets that, w.h.p., for all k ∈ [1 , ε − / ] A L ( L t k ) ⊂ (1 − ( α − δ ) ε ) k L D (4.8)where t k is defined by t k := ε k − X i =0 (1 − ( α − δ ) ε ) i = ε − (1 − ( α − δ ) ε ) k − (1 − ( α − δ ) ε ) . (4.9)Here and in the sequel we assume ε − / to be in N . The value ε − / could equally well bereplaced by any number k f much larger than 1 /ε : the only thing that matters is that t k f is closeto 1 / (2( α − δ )). One remarks that for all values of k (1 − ( α − δ ) ε ) k = r − t k (1 − (1 − ( α − δ ) ε ) ) ε = p − α − δ ) t k + t k O ( ε ) . (4.10) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 15 As ( t k ) k > is bounded above, there exists C > k ∈ [1 , ε − / ] A L ( L t k ) ⊂ (cid:16)p − α − δ ) t k + Cε (cid:17) L D ⊂ ( √ − αt k + η/ L D (4.11)w.h.p. where the second inclusion holds provided that ε and δ are small enough. Combining(4.11), Proposition 4.2 and stochastic coupling, one gets w.h.p. that, for every k ∈ [0 , ε − / ] and t ∈ ( t k , t k +1 ), A L ( L t ) ⊂ ( √ − αt k + (3 η/ L D ⊂ ( √ − αt + η ) L D (4.12)and that, w.h.p., for every t ≥ t ε − / A L ( L t ) ⊂ ( p − αt ε − / + (3 η/ L D ⊂ ηL D . (4.13)This ends the proof of the upper inclusion in (2.17) (note that t ε − / approaches 1 / (2 α ) for ε, δ small). Moreover, (4.13) and stochastic domination implies that, for some constant C , w.h.p. τ + ≤ L α (1 + Cη ) . (4.14)Indeed, it is known from [9] that a droplet of minus spins of linear size ηL disappears within atime τ + which w.h.p. is upper bounded by Cη L .The lower inclusion in (2.17) and the lower bound on τ + are proved in an analogous way using(4.2) instead of (4.1) and (4.4) instead of (4.3). Note that using (4.4) we have to take care tochoose ε small enough but it is possible as t k − t k − is a non-increasing function of k and 1 /ε . (cid:3) Strategy of the proof of Proposition 4.1. Our aim is to use Theorem 3.2 to control themotion of the interface away from the “poles” and Theorem 3.4 (or more precisely Corollary 3.5)to control the motion of the interface close to the “poles”. It is therefore crucial to compare thelocal SSEP or the zero-range dynamics introduced in Sections 3.1 and 3.2 to the true evolutionof the boundary between ”+” and ” − ” spins.As we have already discussed at the beginning of Section 3, however, there exists no exactmapping between the evolution of the height function associated to the two particle processes andthe evolution of the + / − boundary, since the original “ − ” droplet can break into more dropletsand, strictly speaking, the interface cannot be described, even locally, as a height function. Theway out is that, thanks to monotonicity arguments and to the a priori “continuity” informationprovided by Proposition 4.2, we can remove certain updates of the Markov Chain, e.g. freezecertain spins to their initial value. This way, we can show that locally the interface can bestochastically compared to the height function associated to the SSEP (or to the zero-rangeprocess close to the poles). Of course, the detail of the “update removal procedure” is quitedifferent according to whether we want to prove an upper or a lower bound on the “ − domain”.For instance, if we want an upper bound we are allowed to freeze “ − ” spins or to change some“+” into “ − ” spins in the initial condition (this is fine thanks to monotonicity) and at thesame time we can freeze the spins outside (1 + δ ) L D to “+” (this is not allowed directly bymonotonicity, but (4.3) guarantees that such spins stay “+” for all times anyway, w.h.p.). If the“update removal procedure” is performed suitably, the effect is that the various portions of the+ / − interface (away from and close to the poles) then become independent and evolve exactly like the height functions of the SSEP/zero-range process.The approach outlined here will be also used in Section 6 in the case with general convexinitial condition (the generalization of Proposition 4.1 is Proposition 6.2). Upper Bound: proof of (4.1) and (4.3) . The inclusion (4.1) can be rewritten in thefollowing manner, which is more convenient for the proof: for any positive δ , for all ε smallenough, w.h.p. σ x ( εL ) = + for every x ∈ [(1 − ε ( α − δ )) L D ] c . (4.15)Given δ , we fix a value of ξ which is small enough (depending on δ in a way that is specifiedin Section 4.3.2) and set (cf. Figure 6) M ( ε, ξ ) := { ( x, y ) ∈ R , x > ξ and y > ξ } \ [(1 − ε ( α − δ )) D ] N ( ε, ξ ) := { ( x, y ) ∈ R , y > − ξ x ξ } \ [(1 − ε ( α − δ )) D ] . (4.16)Remark that for any ε > M, N and their successive images by rotation of angle π/ π , 3 π/ − ε ( α − δ )) D ] c .PSfrag replacements N ( ε, ξ ) M ( ε, ξ ) ξ ξ − ξ (1 − ε ( α − δ )) D D Figure 6. The light-colored (resp. dark-colored) zones correspond M ( ε, ξ )(resp. N ( ε, ξ )) and its rotations. Together, they form a partition of the comple-ment of (1 − ε ( α − δ )) D (white central region).As the dynamics and the initial shape are invariant under these same rotations, (4.15) isproved if we can show that for ε small enough, w.h.p. σ x ( εL ) = + for every x ∈ LM ( ε, ξ ) (4.17) σ x ( εL ) = + for every x ∈ LN ( ε, ξ ) . (4.18)The above new formulation of (4.1) is very convenient as it allows to consider separately thedynamics close to the poles and away from them.4.3.1. Proof of (4.17) . For any L > 0, we consider the dynamics which has initial condition with” − ” spins in L D and ”+” otherwise, and the same generator as the original dynamics except thatspins on the sites in V := {± } × {− L + , · · · , L − } and on V := {− L + , · · · , L − } × {± } are “frozen to − ”. (The construction of the dynamics is the same as in Section 2.3, except that URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 17 there is no update for these sites). We denote ( σ (1) L ( t )) t ≥ the evolution of this dynamics anddefine A (1) L ( t ) := [ { x : σ (1) x ( t )= − } C x . (4.19)The graphical construction of Section 2.3 gives a natural coupling of σ and σ (1) : A L ( t ) ⊂ A (1) L ( t ) , for every t > . (4.20)The advantage of the “freezing procedure” is that then the evolution in the four quadrantsof ( Z ∗ ) becomes independent. The reason is that the spins on sites ( {± } × Z ∗ ) \ V and( Z ∗ ×{± } ) \ V are “+” for all times (recall that the spins outside the smallest square containingthe initial “ − ” droplet stay “+” forever) so the boundary spins of all four quadrants are frozen.The set A (1) L ( t ) ∩ R is a Young diagram (i.e. a collection of vertical columns of width 1 andnon-negative integer heights, with heights non-increasing from left to right) for all t ≥ ∂ A (1) L ( t ) ∩ R as the graph of a (random) piecewise affine function in thecoordinate system ( f , f ), that we denote by F L ( · , t ). Equation (4.17) is thus proved (for anychoice of ξ ) if one proves that for any ν < − / , for any ε small enough, w.h.p. F L ( x, εL ) ≤ Lf ( x/L, ( α − δ ) ε ) for every x ∈ ( − νL, νL ) , (4.21)where f ( · , t ) is the function whose graph in the coordinate system ( f , f ) is given by the in-tersection of the boundary of (1 − t ) D with the half-plane { ( x, y ) ∈ R , ( x, y ) · f > } (thedomain of definition of f ( · , t ) depends on t but includes [ − − / , − / ] for t small enough). Bydefinition of D , one has f ( · , 0) = f ( · ) (recall the definition of f in (2.12)).In practice, to prove (4.17) one has to prove (4.21) with ν such that 1 / √ − ν = ξ/ √ o ( ξ )for ξ small (with ξ as in (4.16)). The reason is that the point of ∂ D with horizontal coordinate ξ and positive vertical coordinate (in the coordinate system ( e , e )) has horizontal coordinate − (1 − ξ ) / √ o ( ξ ) in the ( f , f ) coordinate system.As explained in Remark 3.1 and on Figure 3, the function F L ( · , t ), up to space rescaling (bya factor √ 2) undergoes the corner flip-dynamics of Theorem 3.2. Thus the scaling limit of F L satisfies the heat-equation or more precisely we have the following convergence in probabilityfor every fixed T > 0: lim L →∞ sup x ∈ [ − √ , √ ] sup t T (cid:12)(cid:12)(cid:12)(cid:12) L F L ( xL, tL ) − g ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 (4.22)where g is the solution for t ≥ x ∈ ( − / √ , / √ 2) of ∂ t g ( x, t ) = ∂ x g ( x, t ) g ( · , 0) = f ( · ) g ( − √ , t ) = g ( √ , t ) = √ . (4.23)Note that the above result plus equation (4.20), plus the fact that g is decreasing in t (sinceit stays concave through time) gives (4.3) of Proposition 4.2 for every t ≤ T < ∞ . Moreover,according to [9, Theorem 1.3], the disappearance time τ + of the minus-droplet is O ( L ) withhigh probability, so that (4.3) also holds for t > T provided that T was chosen large enough. Asa byproduct, we have proven (4.3).Concerning (4.17), in order to prove (4.21) we are reduced to show that for every ν ∈ (0 , / √ x ∈ ( − ν, ν ) , g ( x, ε ) < f ( x, ( α − δ ) ε ) . (4.24) This is a consequence of the way f was determined (see (3.19) and discussion in Section 3.3).First we notice that the time derivative of g is uniformly continuous away from the boundarypoints ± / √ Lemma 4.3. For any < ν < √ , lim t → sup {| ∂ t g ( x, s ) − ∂ t g ( x, | , s ∈ [0 , t ] and x ∈ [ − ν, ν ] } = 0 . (4.25) Proof of Lemma 4.3. This is well known but we sketch a probabilistic proof for the sake ofcompleteness. Let ( B t ) t > denote a standard Brownian motion starting at x ∈ [ − − / , − / ](with the associated expectation denoted by E x ) and let T denote the hitting time of {± / √ } .One has ∂ x g ( x, t ) = E x (cid:2) ∂ x f ( B t ) { t 0) and on (0 , +1) by the definition of D . The boundary condition (3.21) ensures URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 19 continuity of the first derivative of h at zero ( ∂ x h (0) = 0); the reader can check that h hasalso continuous second derivative and that ∂ x h (0) = 12 √ ∂ x f ( − / √ 2) = − α, (4.33)but that the third derivative exhibits a discontinuity in 0.Recall that ξ is the positive constant appearing in (4.16). We set ¯ h : [ − ξ, ξ ] R to be thefunction defined by the following conditions: ¯ h ≡ h on [ − ξ, ξ ], ¯ h is affine on [ − ξ, − ξ ] andon [2 ξ, ξ ] and the derivative ∂ x ¯ h ( · ) is continuous on ( − ξ, ξ ). Since h ( · ) is strictly convex,we have h ( x ) ≤ ¯ h ( x ) with strict inequality outside [ − ξ, ξ ]. Define also the following subsetsof R (cf. Figure 7): J := [4 ξ, ∞ ) × [¯ h (4 ξ ) , ∞ ) J := ( −∞ , − ξ ] × [¯ h (4 ξ ) , ∞ ) . (4.34)To avoid notational complications with integer parts, we assume that L ¯ h (4 ξ ) and 4 Lξ belong to Z ∗ . PSfrag replacements L J L ¯ h (4 ξ ) L J Lξ Lξ LξL ¯ h ( · /L ) L ¯ h ( · /L ) N ( ε, ξ ) Figure 7. In the set L ( J ∪ J ) spins are frozen to ”+” while in the dashedregion they are frozen to “ − ”. The initial condition is “+” in the dark-coloredregion and “ − ” in the light-colored one. The boundary separating dark/lightregion is determined by the function ¯ h ( · ).First of all, observe that, thanks to (4.3), we can freeze the spins in L ( J ∪ J ) to their initialvalue ”+” and, w.h.p., the dynamics will be identical for all times to the original one.Next, we employ a chain of monotonicities, based on the graphical construction of Section2.3. Since we are after an upper bound on the set of minus spins, we can freeze to ” − ” all spinswhose vertical coordinate is below L ¯ h (4 ξ ). Therefore, we have just a dynamics in the set Y := [ − Lξ + 1 , Lξ − × [ L ¯ h (4 ξ ) , ∞ ) . In principle, its initial condition is such that the spin at site ( x , x ) ∈ Y is ” − ” if and onlyif x ∈ [ L ¯ h (4 ξ ) , Lh ( x /L )]. The problem is however that the function x max(¯ h (4 ξ ) , h ( x ))is not concave, which prevents to apply directly Corollary 3.5 later. By monotonicity, we canmodify such initial condition by adding extra ” − ” spins: we therefore stipulate that at time t = 0the spin at site ( x , x ) is ” − ” if and only if x ∈ [ L ¯ h (4 ξ ) , L ¯ h ( x /L )]. Recall that ¯ h ( x ) ≥ h ( x ),so monotonicity goes in the correct direction. With some abuse of notation, we still call ( σ ( t )) t ≥ the dynamics thus modified and A L ( t ) the set of minus spins. We need a final step in orderto map the evolution into the zero-range process. Note that, at time t = 0, the boundary of A L ( t = 0), intersected with the strip [ − Lξ + 1 / , Lξ − / × R , can be identified with thegraph of a c`adl`ag function H L ( · , 0) : [ − Lξ + 1 / , Lξ − / [ L ¯ h (4 ξ ) − / , ∞ ) ∩ Z , which is constant on intervals [ n, n + 1) with n ∈ Z and takes boundary values L ¯ h (4 ξ ) − / H L ( x, 0) is just a discretized version of L ¯ h ( x/L )). However, for time t > A L ( t ) is still the graph of a function, simply becausethe set A L ( t ) can be non-connected (see Figure 4). Let ( σ (2) ( t )) t ≥ be the dynamics obtainedby erasing all the updates that would make A L ( t ) non-connected. It is easy to realize that, since H L ( · , 0) has a single change of monotonicity (from non-decreasing to non-increasing, recall that¯ h ( · ) is concave) such erased updates can only correspond to a ” − ” spin turning into a ”+” spin(see again Figure 4). Therefore, the set of minus spins of the dynamics ( σ (2) ( t )) t ≥ dominatesstochastically A L ( t ): more precisely, we have shown that the coupling given by the graphicalconstruction implies that, w.h.p. and for all t ≥ A L ( t ) ⊂ A (2) L ( t ) := [ { x : σ (2) x ( t )= −} C x . (4.35)We let H L ( · , t ) : [ − Lξ + 1 / , Lξ − / [ L ¯ h (4 ξ ) − / , ∞ ) ∩ Z , denote the piecewise constant (random) function whose graph in the usual coordinates system( e , e ) is the intersection between ∂ A (2) L ( t ) and the strip [ − Lξ + 1 / , Lξ − / × R . Notethat H L ( − Lξ + 1 / , t ) = H L (4 Lξ − / , t ) = L ¯ h (4 ξ ) − / ε small enough, w.h.p.1 L H L ( x, εL ) ≤ h ( x/L, ( α − δ ) ε ) for every x ∈ ( − ξL, ξL ) . (4.36)It is clear from Remark 3.3 that the function H L ( · , · ) follows the dynamics described in Section3.2, with generator (3.8) (here we identify the function H L ( · , t ) with an element of Ω ξL − / , see(3.7)). According to Corollary 3.5 one has for arbitrarily small η > 0, w.h.p, for all x ∈ ( − ξL, ξL )1 L H L ( x, εL ) ≤ ¯ φ ( x/L, (1 + k ∂ x ¯ h k ∞ ) − ε ) + η (4.37)where k ∂ x ¯ h k ∞ = sup [ − ξ, ξ ] | ∂ x ¯ h ( x ) | and ¯ φ ( x, L t ) is the solution of ∂ t ¯ φ ( x, t ) = ∂ x ¯ φ ( x, t )¯ φ ( − ξ, t ) = ¯ φ (4 ξ, t ) = ¯ h (4 ξ )¯ φ ( x, 0) = ¯ h ( x ) for every x ∈ [ − ξ, ξ ] . (4.38)The equation (4.36) is thus proved if one has¯ φ ( x, (1 + k ∂ x ¯ h k ∞ ) − ε ) < h ( x, ( α − δ ) ε ) for every x ∈ [ − ξ, ξ ] . (4.39)Note that by Lemma 4.3 (which is applicable because the second derivative of ¯ h ( · ) = h ( · ) isuniformly continuous in ( − ξ, ξ )) one has, uniformly on [ − ξ, ξ ]¯ φ ( x, (1 + k ∂ x ¯ h k ∞ ) − ε ) = ¯ φ ( x, 0) + ε k ∂ x ¯ h k ∞ ) − ∂ x ¯ φ ( x, 0) + o ( ε )= h ( x ) + 12 (1 + k ∂ x ¯ h k ∞ ) − ( ∂ x h (0) + r ( x )) ε + o ( ε ) (4.40)where r ( x ) tends to 0 for x → 0. Finally, using (4.33), if ξ is chosen small enough so that both r ( x ) and k ∂ x ¯ h k ∞ are sufficiently smaller than δ ,¯ φ ( x, (1 + k ∂ x ¯ h k ∞ ) − ε ) ≤ h ( x ) − ( α − δ/ ε. (4.41) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 21 On the other hand one has h ( x, ( α − δ ) ε ) ≥ h ( x ) − ( α − δ/ ε, (4.42)which ends the proof of (4.18). (cid:3) Lower bound: proof of (4.4) and (4.2) . The proofs follow the same ideas as those ofSection 4.3: we need to control the dynamics for different portions of the interface separately(around the poles and away from them) using the scaling limit results provided by Theorems3.4 and 3.2.4.4.1. Proof of (4.4) . Equation (4.4) is absolutely crucial to start the proof of (4.2) and quiteindependent of the rest. The proof is very similar to that of [2, Theorem 2], so we only sketchthe main steps. Set D := { x ∈ ( Z ∗ ) : d ( x, (1 − δ ) L D ) ≤ } , D ′ := ( Z ∗ ) ∩ (cid:0) (1 + δ ) L D (cid:1) c and consider a modified dynamics (˜ σ ( t )) t ≥ (whose law is denoted ˜ P ), with the same initialcondition as ( σ ( t )) t ≥ and the rules that: (i) after each update, any ” − ” spin which has morethan two ”+” neighbors is turned to ”+”, and the operation is repeated as long as such spinsexist; (ii) the dynamics stops at the time ˜ τ D,D ′ , the first time when there is either a ”+” spinin D or a “ − ” spin in D ′ . We define also ˜ τ D the first time when there is a “+” spin in D and τ D,D ′ , τ D the analogous random times for the original dynamics.Note that, by (4.3), w.h.p. τ D,D ′ = τ D . Note also that the two dynamics can be coupled in away that τ D,D ′ = τ D implies ˜ τ D,D ′ = ˜ τ D ≤ τ D (thanks to point (i) above, since before ˜ τ D,D ′ themodified dynamics has less “ − ” spins that the original one). Therefore, P ( τ D ≤ εL ) = P ( τ D ≤ εL ; τ D,D ′ = τ D ) + o (1) ≤ ˜ P (˜ τ D ≤ εL ; ˜ τ D,D ′ = ˜ τ D ) + o (1)and it suffices to prove for instance that˜ P (˜ τ D,D ′ ≤ εL ; ˜ τ D = ˜ τ D,D ′ ) ≤ exp( − γL )for some ε = ε ( δ ) > , γ > 0. For this, one first observes (as in [2, Eq. (8.6)]) that when˜ τ D,D ′ = ˜ τ D the difference between the number of “+” spins at time ˜ τ D and the number of “+”spins at time 0 is at least cδ L deterministically, for some c > P ( |{ x : ˜ σ x ( εL ) = + }| − |{ x : ˜ σ x (0) = + }| ≥ cδ L ) ≤ exp( − γL )if ε = ε ( δ ) is small enough. This is based on the fact (cf. [2, Lemma 8.5]) that, for times smallerthan ˜ τ D,D ′ , the rate of increase of the number of ”+” spins is uniformly bounded by a constant.4.4.2. Scheme of the proof of (4.2) . Given some fixed δ > 0, we want to prove that for ε > − ( α + δ ) ε ) L D ⊂ A L ( εL ) , (4.43)or equivalently σ x ( εL ) = − for every x ∈ (1 − ( α + δ ) ε ) L D . (4.44)Given ξ small enough (depending on δ ) and ν small enough (depending on ξ ), we define (cf.Figure 8) U := (1 − ν ) D ,A ( ε ) := [((1 − ( α + δ ) ε ) D ) \ U ] ∩ [ ξ, + ∞ ) ,B ( ε ) := [((1 − ( α + δ ) ε ) D ) \ U ] ∩ ([ − ξ, ξ ] × R + ) . (4.45)and A i , B i , i = 2 , , A ( ε ) , B ( ε ) by the rotation of angle ( i − π . PSfrag replacements A B Uξ ξ ξ D Figure 8. The large droplet is D and (1 − ε ( α + δ )) D is obtained by removingthe external dark layer. The white central region U , together with A , B andits rotations (deformed rectangular regions) form a partition of (1 − ε ( α + δ )) D .One has (1 − ( α + δ ) ε ) D = U ∪ [ i =1 A i ! ∪ [ i =1 B i ! , (4.46)and hence (using rotational symmetries), to prove (4.43), it is sufficient to prove that for ε smallenough, w.h.p. L U ⊂ A L ( εL ) , (4.47) LA ( ε ) ⊂ A L ( εL ) , (4.48) LB ( ε ) ⊂ A L ( εL ) . (4.49)The first line, i.e. Equation (4.47), is a direct consequence of (4.4) provided that ε is chosensmall enough (how small depending on ν ). Actually, one has the following stronger statementthat will be useful for what follows: if ε is small then w.h.p. L U ⊂ A L ( tL ) for every t ≤ ε. (4.50)The main work is thus to prove (4.48) and (4.49).4.4.3. Proof of (4.49) . This is similar to the proof of (4.18), except that monotonicities will beneeded in the opposite direction.Let ¯ h : [ − ξ, ξ ] R be a concave, twice differentiable, even function such that¯ h ( x ) = h ( x ) , ∀ x ∈ [ − ξ, ξ ] , ¯ h ( x ) < h ( x ) , ∀ x ∈ [ − ξ, − ξ ) ∪ ( ξ, ξ ] (4.51) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 23 where h ( · ) was defined in Section 4.3.2 to be the graph of ∂ D ∩ ( R × R + ) in the ( e , e )coordinate system. Once ξ is fixed, we choose ν and ¯ h such that the point (2 ξ, ¯ h (2 ξ )) lies in theinterior of U .Using equation (4.50), we can freeze the spins with vertical coordinate L ¯ h (2 ξ ) and horizontalcoordinate in ( − Lξ, Lξ ) (we assume for notational convenience that 2 Lξ and L ¯ h (2 ξ ) are in Z ∗ ) to their initial value ” − ”, and w.h.p. , the dynamics we obtain is identical to the originalone up to time εL . PSfrag replacements A B L UL ¯ h ( · /L ) Lξ LξL D Figure 9. The sites in the dashed vertical lines are frozen to ”+” and those ofthe horizontal bold segment to ” − ” so that the dynamics in the colored infiniterectangle is independent of the rest of the system. At time t = 0 the sites in thedark-colored region (whose upper boundary is determined by ¯ h ( · ) are ” − ” whilethose of the light-colored one are ”+”. The function ¯ h ( · ) is such that the base ofthe dark-colored region is in LU .Next we use a chain of monotonicities based on the graphical construction of Section 2.3.Since we are after a lower bound on the set of minuses, we can freeze to ”+” all the spins withhorizontal coordinate ± Lξ and vertical coordinate larger than L ¯ h (2 ξ ). Once this is done, weare reduced to considering the dynamics restricted to the set Y := [ − Lξ + 1 , Lξ − × [ L ¯ h (2 ξ ) + 1 , ∞ ) , (4.52)as spins on its boundary are fixed. In principle, the initial condition one should consider is suchthat ( x , x ) ∈ Y has spin ” − ” iff x ∈ [ L ¯ h (2 ξ ) + 1 , Lh ( x /L )], but again by monotonicity,we can add extra ”+” spins: we stipulate that, at time t = 0, ( x , x ) has spin ” − ” iff x ∈ [ L ¯ h (2 ξ ) + 1 , L ¯ h ( x /L )]. With some abuse of notation, the dynamics thus modified is still called( σ ( t )) t ≥ .As for the proof of (4.18), we need a final step to map the dynamics onto the interfacedynamics of Theorem 3.4, the problem being exactly the same as then: it is not true thatthe boundary of A L ( t ) stays connected for all t . The solution adopted in the previous section(leading to the dynamics ( σ (2) ( t )) t , see discussion before (4.35)) does not work here as we arenow looking for a lower bound .Let ( σ (3) ( t )) t be the dynamics that evolves like ( σ ( t )) t except that any spin that has three ”+”neighbors is turned instantaneously to ”+” (see Figure 10). The coupling given by graphicalconstruction implies that [ { x : σ (3) x ( t )= −} C x =: A (3) L ( t ) ⊂ A L ( t ) . (4.53) Moreover our choice of initial condition guarantees that A (3) L ( t ) stays connected for all time,since the set D is convex. PSfrag replacements A B Figure 10. Light-colored (resp. dark-colored) squares denote “ − ” (resp. “+”)spins. In our modified dynamics σ (3) , when a spin has three “+” neighbors, itis instantaneously turned to “+”. On the figure, if spin at A is updated andturns to “+”, then the spin B has three “+” neighbors and therefore also turnsinstantaneously to “+”.We denote by H L ( · , t ) the c`adl`ag function [ − ξ, ξ ] R whose graph corresponds to theintersection between ∂ A (3) L ( t ) and the vertical strip [ − Lξ + 1 / , Lξ − / × R . Note that H L ( · , t ) can be visualized as a collection of columns of width 1 and integer height. With thisnotation and (4.53), equation (4.49) is proved if one has w.h.p.1 L H L ( x, L ε ) ≥ h ( x/L, ( α + δ ) ε ) for every x ∈ ( − ξL, ξL ) . (4.54)Now we want to relate the dynamics of H L to that of Theorem 3.4. The relation is almostidentical to that discussed in Remark 3.3, except for a slight difference in the way particles oftypes A and B annihilate in the zero-range process. Given Z ∋ x = − Lξ + 1 / , . . . , Lξ − / n > t at site x if lim y → x + H L ( y, t ) − lim y → x − H L ( x, t ) = n and that there are n > − n . Then it is easy to realize that, under the dynamics ( σ (3) ( t )) t > , each particleperforms a symmetric simple random walk with jump rate 1 / (2 n ) both to right or left (with n the occupation number of the site where the particle is), and that particles of different typeannihilate immediately if they are at sites of distance on the same site ): this isthe effect of flipping instantaneously ” − ” spins with more than two ”+” neighbors. Note alsothat, due to convexity of ¯ h ( · ), particles of type A are always to the left of particles of type B.Therefore, if we take H L ( · , t ) and we eliminate one of the columns of maximal height (see Figure11) (note that there are always at least two), the modified height function thus obtained followsexactly the evolution of Theorem 3.4.Of course, the erased column does not change the scaling limit so that one can apply Theorem3.4 and Corollary 3.5 and get that for any t and η > 0, w.h.p.1 L H L ( x, L t ) ≥ ¯ φ ( x/L, t ) − η for every x ∈ {− ξL, . . . , ξL } , (4.55)where ∂ t ¯ φ ( x, t ) = ∂ x ¯ φ ( x, t )¯ φ (2 ξ, t ) = ¯ φ ( − ξ, t ) = ¯ h (2 ξ )¯ φ ( x, 0) = ¯ h ( x ) for every x ∈ [ − ξ, ξ ] . (4.56) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 25 PSfrag replacements Supressed column0 0 N N − Figure 11. Left: the height function associated to the “+ / − ” boundary forthe dynamics σ (3) ( t ). Right: the same height function, with one of the highestcolumns removed; this follows the same evolution as in Theorem 3.4. The factthat the new interface is step shorter makes no difference in the macroscopiclimit.Therefore, (4.54) is proved if one can check that¯ φ ( x, ε ) > h ( x, ( α + δ ) ε ) for every x ∈ [ − ξ, ξ ] . (4.57)The above equation is proved is the same manner as (4.39): one just needs to choose ξ smallenough. (cid:3) Proof of (4.48) . First of all, one freezes to ” − ” all the spins on the cross-shaped regionof sites in L U (cf. (4.45)) such that at least one of their coordinates is ± / 2. Equation (4.50)guarantees that if ε is chosen small enough, w.h.p. the so-obtained dynamics coincides with theoriginal one up to time εL if ε is small enough.Then one defines ( σ (4) ( t )) t > as the dynamics obtained by changing the initial condition inthe following manner: all spins ( x, y ) ∈ L D with either | x | > L (1 − ν ) or | y | > L (1 − ν ) arechanged from ” − ” to ”+” (recall that ν is the constant that enters the definition (4.45) of U )and therefore they stay “+” forever, since they have at least three “+” neighbors. Note that,this way, the evolution in each quadrant of ( Z ∗ ) is independent. By monotonicity, we get thatw.h.p, for every t ≤ εL , [ { x : σ (4) x ( t )= −} C x =: A (4) L ( t ) ⊂ A L ( t ) . (4.58)and therefore (4.48) is proved if one can show that LA ( ε ) ⊂ A (4) L ( εL ) . (4.59)Next, note that ∂ A (4) L ( t ) ∩ R in the coordinate system ( f , f ) is the graph of a randompiecewise affine function F L : (cid:20) − (1 − ν ) L √ , (1 − ν ) L √ (cid:21) R which undergoes the corner-flip dynamics described in Theorem 3.2 (apart from space rescalingby a factor √ L →∞ sup x ∈ [ − − ν √ , − ν √ ] sup t ε (cid:12)(cid:12)(cid:12)(cid:12) L F L ( xL, tL ) − g ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 (4.60) where ∂ t g ( x, t ) = ∂ x g ( x, t ) g ( − − ν √ , t ) = g ( − ν √ , t ) = − ν √ g ( x, 0) = ¯ f ( x ) for every x ∈ h − − ν √ , − ν √ i (4.61)and ¯ f is the profile of the initial condition, i.e.¯ f ( x ) := min( f ( x ) , (1 − ν ) √ − | x | )) . (4.62)Let P (resp. P ) be the point on ∂ D whose coordinates ( x, y ) (resp. ( x , y )) in the coordinatesystem ( e , e ) satisfy x > , y = 1 − ν (resp. x > , y = h ( ξ )). Call − d < − d < P (resp. of P ) in the coordinate system ( f , f ) .In view of (4.60) and of Definition (4.45) of A ( ε ), equation (4.59) is satisfied if g ( x, ε ) > f ( x, ( α + δ ) ε ) for every x ∈ ( − d , d ) , (4.63)The proof of this is very similar to that of (4.24) provided that ¯ f coincides with f in a domaincontaining strictly ( − d , d ) (this guarantees for instance that ∂ x ¯ f ( · ) is uniformly continuous ina domain containing ( − d , d ), so that the drift ∂ t g is continuous in time, cf. Lemma 4.3). Forthis to hold, it is enough to assume that d > d , i.e. that ν in (4.45) has been chosen sufficientlysmall as a function of ξ so that 1 − ν > h ( ξ ). (cid:3) Proof of Theorem 2.1: existence of anisotropic curve-shortening flow withconvex initial condition Let us first recall some properties of the support function h ( · ) of a convex curve γ . First ofall, if γ is contained in the convex set delimited by γ ′ then h ( θ ) ≤ h ′ ( θ ) for every θ . Next, thesupport function is related to the curvature and to the length L ( γ ) of γ by (cf. [12, Lemma 1.1]) ∂ θ h ( θ ) + h ( θ ) = 1 k ( θ ) (5.1) L ( γ ) = Z π h ( θ )d θ = Z π k ( θ ) d θ. (5.2)Also (cf. Lemma 4.1.1 in [13], with the warning that what they call θ is θ − π/ x ( θ ) , y ( θ )) of the point of γ where the outward directed normal forms ananticlockwise angle θ with the positive horizontal axis can be expressed as x ( θ ) = h (0) − Z θ sin( s ) k ( s ) ds (5.3) y ( θ ) = h ( π/ 2) + Z θπ/ cos( s ) k ( s ) ds. (5.4)Under the flow (2.7), the time derivatives of area and length are (cf. [12, Lemma 2.1]) ddt Area ( γ ( t )) = − Z π a ( θ )d θ (5.5) ddt L ( γ ( t )) = − Z π a ( θ ) k ( θ, t )d θ. (5.6)For the moment these are formal statements since we do not know yet that the flow exists. URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 27 Proof of Theorem 2.1. Uniqueness of the flow is trivial, so we concentrate on existence.First of all, we need to regularize the functions a ( · ) and k ( · ). Given 0 < w < a ( w ) ( · )to be a family of smooth approximations of the anisotropy function a ( · ). More precisely: Assumption 5.1. (1) a ( w ) ( · ) is π -periodic and C ∞ ;(2) a ( w ) ( θ ) w → −→ a ( θ ) uniformly in θ ;(3) for fixed θ , the function w a ( w ) ( θ ) is non-increasing;(4) the function a ( w ) ( · ) is Lipschitz, uniformly in w > (this is possible because the function a ( · ) itself is − Lipschitz);(5) the functions w 7→ k ∂ θ a ( w ) k ∞ := max θ | ∂ θ a ( w ) ( θ ) | and w 7→ k ∂ θ a ( w ) k ∞ are bounded,uniformly for w in any compact subset of (0 , . A possible choice is a ( w ) ( θ ) = ( a ∗ g ( w ) )( θ ) + ε w where g ( w ) is a centered Gaussian of variance w . In the convolution it is understood that a ( · )is seen as a 2 π -periodic function on R and ε w is chosen so that a ( w ) ( · ) satisfies the monotonicitywith respect to w . It is easy to check that one can choose ε w = − Cw for some suitably large C .Indeed, monotonicity in w is guaranteed if for w ′ < w one has ε w ′ − ε w ≥ k a ∗ ( g ( w ) − g ( w ′ ) ) k ∞ . On the other hand, since a ( · ) is Lipschitz, one sees easily that k a ∗ ( g ( w ) − g ( w ′ ) ) k ∞ = O ( w − w ′ ).Also, we approximate γ with a sequence of convex curves ( γ ( w ) ) 0) = h ( w ) ( θ ) (5.7)admits a solution corresponding to a flow of curves ( γ ( w ) ( t )) t ≥ which remain convex and shrinkto a point in a finite time ˜ t f := t ( w ) f = Area ( γ ( w ) (0)) / Z π a ( w ) ( θ )d θ (cf. (5.5) with a ( · ) replaced by a ( w ) ( · )). For lightness of notation, we will often write ˜ h ( · , · ) , ˜ γ ( t ) , ˜ a ( · ),etc. for the regularized quantities h ( w ) ( · , · ) , γ ( w ) ( t ) , a ( w ) ( · ), etc. Thanks to Assumption 5.1, we have that R π a ( w ) ( θ )d θ → R π a ( θ )d θ = 2 as w → t ( w ) f = t f (1 + o (1)) when w → 0, with t f defined in Theorem 2.1.From (5.1) and (5.7) one can check that the curvature satisfies the parabolic equation ( ∂ t ˜ k = ˜ k ∂ θ (˜ a ˜ k ) + ˜ a ˜ k ˜ k ( θ, 0) = ˜ k ( θ ) . (5.8)Also, following [13] it is possible to see that the curvature function stays C ∞ until ˜ t f (since ˜ a is C ∞ ). However, estimates on the regularity will not be necessarily uniform in the regularizationparameter w and we will need to be very careful on this point.For fixed t , set γ ( t ) := lim w → γ ( w ) ( t ) (5.9)where convergence is in the Hausdorff metric. A posteriori , since we will see that ( γ ( t )) t providesthe (unique) solution to our curve-shortening equation, it follows that the limit (5.9) does notdepend on the choice of regularization. Existence of the limit (in the Hausdorff metric) alongsub-sequences is guaranteed by the Blaschke selection theorem [8, Th. 32] which says that afamily of convex subsets of a bounded subset of R n admits a sub-sequence converging to a non-empty convex set. Uniqueness of the limit follows from the fact that γ ( w ′ ) ( t ) ⊂ γ ( w ) ( t ) if w ′ < w and t < t ( w ′ ) f (because a ( w ) ( θ ) is decreasing in w and that the curve is smooth at all times). Onehas to use Convergence in Hausdorff distance also holds for the boundary curves.Since the volume is continuous in the topology induced by the Hausdorff metric [8, Ch. 4]we also see that Area ( γ ( t )) = Area ( γ ) − t R π a ( θ )d θ = Area ( γ ) − t ; for t → t f the curve γ ( t )shrinks to a point (its diameter shrinks to zero). We will prove Theorem 5.1. The flow of curves ( γ ( t )) t For t < ˜ t f let ˜ k max ( t ) (resp. ˜ k min ( t ) ) be the maximal (resp. minimal) curvatureof ˜ γ ( t ) . We let ˜ k max := ˜ k max (0) and similarly for ˜ k min and ˜ a max(min) := max θ (min θ )˜ a ( θ ) . Also, k min(max) and a min(max) are defined similarly to ˜ k min(max) , ˜ a min(max) but with ˜ k ( · ) , ˜ a ( · ) replaced by k ( · ) , a ( · ) . It is crucial that ˜ k max ( t ) stays bounded, uniformly for w small, as long as the disappearancetime is not approached: Proposition 5.3 (Regularity estimate) . Assume that the curvature function k ( · ) is Lipschitz.There exists w > such that, for every b > , t < t f (1 − b ) , < w ≤ w one has ˜ k max ( t ) ≤ C (5.10) and max θ | ∂ θ (˜ a ( θ )˜ k ( θ, t )) | ≤ C ( L ( k ) + 1) (5.11) where we recall that L ( k ) is the Lipschitz constant of the function k ( · ) . The constants C and C depend only on b and on k max .Proof of Proposition 5.3. The proof is based on ideas of [12]. However, it is important to makesure that estimates are uniform in w ≤ w (in [12] the anisotropy function a ( · ) is assumed to be C , so there was no need to regularize it). URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 29 Fix w > 0. First we get a lower bound on ˜ k min ( t ). Note first of all that at time zero theminimal curvature is bounded away from zero (uniformly in w ): indeed, using (5.2) and the factthat the curvature function is L ( k )-Lipschitz, L ( γ (0)) = Z π k ( θ ) d θ ≥ Z π k min + L ( k ) θ d θ = 2 L ( k ) log L ( k ) π + k min k min . (5.12)Then, since the length of γ (0) is finite, k min must be positive.Set for simplicity g = g ( θ, t ) = ˜ a ( θ )˜ k ( θ, t ) . Formula (5.8) gives ∂ t g = 1˜ a (cid:0) g ∂ θ g + g (cid:1) =: g ( θ, t ) u ( θ, t ) . (5.13)This, together with the fact that ˜ a ( · ) and ˜ k ( · , t ) are smooth, implies that ddt min θ g ≥ min θ g ˜ a max ≥ k min ( t ) ≥ ˜ a min ˜ a max ˜ k min ≥ Ck min > C independent of w (say for w ≤ w ) thanks to the uniform convergence a ( w ) ( · ) → a ( · )and k ( w ) ( · ) → k ( · ).Next the real work: bounding ˜ k max ( t ) uniformly in w . From (5.13) one sees that, since ˜ a ( · )and ˜ k ( · , t ) are smooth, ddt max θ g ≤ a min × (max θ g ) . (5.16)From this one immediately gets that ˜ k max ( t ) is upper bounded uniformly in w ≤ w , up to sometime t depending only on k max . However the solution of ˙ x = x explodes in finite time, andcertainly before the time ˜ t f when the curve shrinks to a point, so we need to do better.For this, we define z ( t ) = min θ u ( θ, t ) (cf. (5.13)). Then, taking the derivative of u withrespect to t shows (cf. Lemma 4.2 of [12] for details) thatdd t z ( t ) ≥ z ( t ) so that if z (0) ≥ z ( t ) ≥ 0, if z (0) ≤ z ( t ) ≥ − / | z (0) | + 2 t ). Altogether, weget that u ( θ, t ) ≥ − t (5.17)uniformly in θ and w ≤ w . Now we use this to get a uniform bound on k ∂ θ g k ∞ in terms of˜ k max ( t ). Without loss of generality suppose that there exists θ such that ∂ θ g ( θ , t ) = k ∂ θ g k ∞ (if this is not the case one can still find θ such that ∂ θ g ( θ , t ) = −k ∂ θ g k ∞ and apply the samemethod). Let also θ > θ be such that ∂ θ g ( θ , t ) = 0 (such an angle exists since g is periodic). Then, from the definition (5.13) of u , k ∂ θ g k ∞ = − Z θ θ ∂ θ g d θ = − Z θ θ (cid:18) u ( θ, t )˜ k ( θ, t ) − ˜ a ( θ, t )˜ k ( θ, t ) (cid:19) d θ ≤ t Z θ θ d θ ˜ k ( θ, t ) + ( θ − θ )˜ a max ˜ k max ( t ) ≤ L ( γ (0)) t + C ˜ k max ( t ) . (5.18)In the last inequality we used (5.2) and then (5.6) which says that L (˜ γ ( t )) ≤ L (˜ γ (0)) ≤ L ( γ (0)).Since g = ˜ a ˜ k and by assumption ˜ a is C ∞ and Lipschitz uniformly in w , one deduces that k ∂ θ ˜ k k ∞ ≤ L ( γ (0)) t + C ˜ k max ( t ) ≤ C ( t )˜ k max ( t ) (5.19)and C can be chosen to be decreasing in t . From this it is trivial to see that, if θ is such that˜ k ( θ , t ) = ˜ k max ( t ), one has˜ k ( θ, t ) ≥ ˜ k max ( t ) / | θ − θ | ≤ α ( t ) (5.20)for some α ( t ) increasing in t (it could vanish for t → t < (1 − b ) t f E ( t ) := Z π ˜ a ( θ ) log( g ( θ, t ))d θ ≤ C (5.21)where C depends only on a max and on b and on the maximal curvature k max of the initial curve γ (0). Indeed, (5.21) is obvious for t = 0, since the initial curvature is bounded by assumption.To get the control for t > 0, one observes (cf. Propositions 5.3 and 5.4 of [12]) thatdd t E ( t ) ≤ a max L (˜ γ (0)) Area (˜ γ ((1 − b ) t f )) (cid:18) − dd t L (˜ γ ( t )) (cid:19) . The prefactor is bounded since b > L (˜ γ (0)). At this point we are almost done: using (5.20) C ≥ Z π ˜ a ( θ ) log( g ( θ, t ))d θ (5.22) ≥ α ( t )˜ a min log(˜ a min ˜ k max ( t ) / 2) + 2 π ˜ a max log[min(1 , ˜ a min ˜ k min ( t ))] (5.23)and this (recall that ˜ k min ( t ) ≥ Ck min > 0, cf. (5.15)) gives us an upper bound on ˜ k max ( t )uniformly in w ≤ w and t < (1 − b ) t f : up to t one uses the upper bound which comesfrom (5.16) and after t the one from (5.22); Eq. (5.10) is proven. When t approaches thedisappearance time ˜ t f (i.e. when b approaches zero) the upper bound diverges (because C diverges), as it should.Equation (5.19) says that the curvature function is Lipschitz with a Lipschitz constant C thatdepends on t, b and L (0) but not on w . This is not yet the desired (5.11) because the bounddiverges for t → 0. To prove (5.11) remark that, using (5.13), ∂ t ∂ θ g = ∂ θ (cid:18) g ˜ a ∂ θ g + g ˜ a (cid:19) = − ∂ θ ˜ a ˜ a (cid:0) g ∂ θ g + g (cid:1) (5.24)+ 1˜ a (cid:0) g∂ θ g∂ θ g + g ∂ θ g + 3 g ∂ θ g (cid:1) . (5.25)At the point where ∂ θ g is maximized, ∂ θ g cancels and ∂ θ g is non-positive. This, together withthe boundedness of g uniformly in w ≤ w , θ ∈ [0 , π ] and t < (1 − b ) t f , implies ∂ t max θ ∂ θ g ( θ, t ) ≤ C (1 + max θ ∂ θ g ( θ, t )) . (5.26) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 31 where C just depends on k max and b . Integrating with respect to time, one getsmax θ ∂ θ g ( θ, t ) ≤ C (cid:20) max θ ∂ θ ( a ( w ) ( θ ) k ( w ) ( θ )) + 1 (cid:21) with C depending only on C . Also, observe that ∂ θ ( a ( w ) ( θ ) k ( w ) ( θ )) ≤ (3 / | ∂ θ k ( w ) ( θ ) | + C ≤ (3 / L ( k ( w ) ) + C with C a constant depending on k max , since for w small a ( w )max < (3 / 4) and a ( w ) is uniformlyLipschitz. Finally, from Assumption 5.2 (3), we can conclude ∂ θ ( a ( w ) ( θ ) k ( w ) ( θ )) ≤ C + L ( k )for w small. An analogous lower bound can be found on ∂ t min θ ∂ θ g ( θ, t ) and this gives (5.11). (cid:3) Following [13] it is possible to prove that, once we have bounds on the curvature and on k ∂ θ g ( · , t ) k ∞ , for every n ≥ t < t f (1 − b ) the derivatives ∂ nθ g ( θ, t ) are also bounded. Thebounds we get are in general not uniform in w but this is not very important for our purposes.Indeed, we will need only: Proposition 5.4. Fix b > . There exists a function c ( w ) , which is non-increasing with respectto w ∈ (0 , w ] such that for t < (1 − b ) t f max θ | ∂ t ˜ h ( θ, t ) | ≤ c ( w ) . (5.27) Proof. Recall (5.7) and (5.13): ∂ t ˜ h = − a ( g ∂ θ g + g ) . (5.28)Thus we just have to bound ∂ θ g , since we have already proved that g itself is bounded. For this,we adapt the method used by Gage and Hamilton in [13] for the special case of the isotropiccurve shortening flow where a ≡ 1. What they observed [13, Lemma 4.4.2] is that, if thecurvature and its θ -derivative are bounded (which we proved in Proposition 5.3), the t -derivativeof Φ( t ) := R π [ ∂ θ g ( θ, t )] d θ can be upper bounded by a constant times Φ( t ) itself and then onecan integrate the inequality with respect to t to get a bound on Φ( t ) in terms of Φ(0). Inour case, with a similar computation, we find that ( d/dt )Φ( t ) is upper bounded by Φ( t ) timesa constant depending on k ∂ θ a ( w ) k ∞ , which is finite uniformly for w ≤ 1. Since Φ(0) is alsobounded for w in compact subsets of (0 , 1) (cf. Assumption 5.1 (5) and Assumption 5.2 (4)), weget that Φ( t ) ≤ c ( w ) for w ∈ (0 , 1) and t < (1 − b ) t f and we can choose c to be decreasing. Ingeneral, c will diverges when w approaches zero.A similar computation (cf. [13, Lemma 4.4.3] when a ( θ ) ≡ 1) shows thatΨ( t ) := Z π [ ∂ θ g ( θ, t )] d θ ≤ c ( w )with c ( · ) decreasing in w ∈ (0 , π -periodicfunction f one has (cf. [13, Corollary 4.4.4]) k f k ∞ ≤ C Z π ( f + ( ∂ θ f ) )d θ for some universal constant C , applied with f ( · ) = ∂ θ g ( · , t ), to get that k ∂ θ g k ∞ ≤ c ( w ) as wewished. (cid:3) Proof of Theorem 5.1. We are now ready to prove that ( γ ( t )) t provides a classical solution of(2.7). This is based on the following easy consequence of the Arzel`a-Ascoli Theorem: Lemma 5.5. Let f ( n ) be a sequence of periodic C functions on [0 , π ] , such that both sequences f ( n ) and ∂ x f ( n ) are uniformly bounded and equicontinuous. If f ( n ) → f as n → ∞ , then f is C and ∂ x f = lim n ∂ x f ( n ) , where the convergence is uniform and does not require sub-sequences. First of all, we note that ˜ h ( · , t ) does converge (for w → 0) to h ( · , t ) for every fixed t < t f . Thisjust follows from the fact that ˜ γ ( t ) converges to γ ( t ) in terms of Hausdorff distance. Furthermore,convergence is uniform in t < t f (1 − b ) for every fixed b . This is true because the area differencebetween ˜ γ ( t ) and γ ( t ) ⊂ ˜ γ ( t ) is small with w (uniformly in t ) and the curvature is uniformlybounded: then, if ˜ h ( θ, t ) − h ( θ, t ) where larger than some δ independent of w for some ( θ, t ),necessarily the area difference would be larger than some c ( δ ) at that time.Applying Lemma 5.5 and recalling (5.1), we get that, for t fixed, ∂ θ ˜ h ( θ, t ) and ˜ k ( θ, t ) con-verge to ∂ θ h ( θ, t ) and k ( t, θ ) respectively and that convergences are uniform in θ (knowing thatthe curvature is Lipschitz is important here). Note by the way that k ( · , t ) is Lipschitz, since k ∂ θ ˜ k ( · , t ) k ∞ is uniformly bounded.Then applying dominated convergence (which is allowed in view of Proposition 5.3), one getsthat h ( θ, t ) − h ( θ, s ) = − Z ts a ( θ ) k ( θ, u )d u, (5.29)which is an integrated version of (2.7). To get the stronger statement (2.7), we need to provethat k ( θ, t ) is continuous as a function of t .First of all, we prove that one can find a function ε : (0 , ∋ w ε ( w ) ∈ R + , increasing andgoing to zero as w → θ , for all t ≤ (1 − b ) t f , | ˜ k ( θ, t ) − k ( θ, t ) | ≤ ε ( w ) . (5.30)If this were not the case then, thanks to the fact that ˜ k ( · , t ) and k ( · , t ) are uniformly Lipschitz,we would have, say, for arbitrarily small w and for some ε > t < t f (1 − b ),˜ k ( θ, t ) − k ( θ, t ) ≥ ε for θ ∈ [¯ θ, ¯ θ + ε ] for some ¯ θ ∈ [0 , π ]. But then, since (cf. (5.1))( ∂ θ + 1)( h ( θ, t ) − ˜ h ( θ, t )) = 1 k ( θ, t ) − k ( θ, t ) , this would contradict the uniform convergence of ˜ h ( · , · ) to h ( · , · ).On the other hand, from Proposition 5.4, for all θ and for all t, s ≤ (1 − b ) t f | ˜ k ( θ, t ) − ˜ k ( θ, s ) | ≤ c ( w ) | t − s | . (5.31)Together with (5.30) this implies that | k ( θ, t ) − k ( θ, s ) | ≤ inf w (2 ε ( w ) + c ( w ) | t − s | ) . (5.32)The right-hand side clearly tends to zero with | t − s | (choose a sequence { w k } tending to zero. If c ( w k ) does not diverge we are done. Otherwise, compute the right-hand side for the w = w k withthe largest value of k such that c ( w k ) ≤ | t − s | − / ). This shows that t k ( θ, t ) is continuousaway from t f and the proof is complete. (cid:3) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 33 Proof of Theorem 2.2: evolution of a convex droplet The proof is very similar to that of Theorem 2.3 in the scale-invariant case (Section 4), andtherefore it will be only sketched. We will also try to use as much as possible the same notationsas in Section 4.First we present two statements that are analogous to Propositions 4.2 and 4.1: Proposition 6.1. Let D be convex with a bounded curvature function. For every α > , w.h.p. A L ( L t ) ⊂ L D ( α ) for every t ≥ (recall definition (2.8) ). Moreover, for every α > there exists ε ( α, k max ) > such that w.h.p A L ( L t ) ⊃ L D ( − α ) for every t ∈ [0 , ε ] . (6.2) Proof. The proof of (6.1) is essentially identical to that of (4.3), so we give no detail. As for(6.2), given α it is possible to give a finite collection { D i } i such that: • each D i is an open convex subset of R , obtained from (the interior of) the invariantshape D via a suitable translation and shrinking; • D i ⊂ D for every i ; • ∪ i D i ⊃ D ( − α/ .Given η > 0, thanks to Proposition 4.2 there exists ε > t < ε onehas A L ( L t ) ⊃ ∪ i (cid:16) L D ( − η ) i (cid:17) . Here we use monotonicity (because D i ⊂ D ) and the fact that the union of a finite number ofevents which occur w.h.p. still has probability tending to 1. Note that the choice of ε is dependingon η but also on the diameter of the smallest set in the collection { D i } i and consequently on k max . Then, if η is small enough (depending on α ) it is clear that ∪ i D ( − η ) i ⊃ D ( − α ) (recall thatthe D i are open sets, so that every x ∈ D ( − α ) is contained in the interior of at least one D i ). (cid:3) Proposition 6.2. Let D be convex whose curvature function is L ( k ) -Lipschitz and is boundedaway from zero and infinity. For all δ > there exists ε ( δ, k min , k max , L ( k )) > such that forall < ε < ε , w.h.p., A L ( L ε ) ⊂ L D ( ε (1 − δ )) , (6.3) and A L ( L ε ) ⊃ L D ( ε (1 + δ )) (6.4) where we recall that D ( t ) is the set enclosed by the curve γ ( t ) .Proof of Theorem 2.2 assuming Propositions 6.2 and 6.1. It is enough to prove (2.9) for t < (1 − b ) t f for arbitrary b > 0. Then, the statement for t ≥ (1 − b ) t f and also (2.11) follows fromthe fact that the disappearence time of a droplet of diameter ℓ is w.h.p. O ( ℓ ) (recall that γ ( t )shrinks to a point when t → t f in the sense that its diameter converges to zero). Define k ∗ min > k ∗ max , L ∗ k < ∞ ) to be the infimum (resp. maximum) of k min ( s ) (resp. k max ( s ) , L ( k ( s )))on [0 , (1 − b ) t f ]. Fix δ ′ small and let ε < ε ( δ ′ , k ∗ min , k ∗ max , L ∗ k ) and ε < ε ( δ/ , k ∗ max ) with ε , ε defined in Propositions 6.1 and 6.2. Using the Markov property and the monotonicity of ourprocess we get that, w.h.p., for any k such that εk < (1 − b ) t f A L ( L kε ) ⊂ L D ( kε (1 − δ ′ )) . (6.5)From (6.5) and Proposition 6.1 we get that w.h.p., for every t (1 − b ) t f , A L ( L t ) ⊂ L (cid:20) D (cid:18)(cid:22) tε (cid:23) ε (1 − δ ′ ) (cid:19)(cid:21) ( δ/ ⊂ L (cid:2) D (( t − ε )(1 − δ ′ )) (cid:3) ( δ/ . (6.6) Setting ε ′ = t f δ ′ + ε this implies that w.h.p. A L ( L t ) ⊂ L (cid:2) D ( t − ε ′ ) (cid:3) ( δ/ for every t (1 − b ) t f . (6.7)Finally observe (this follows from (2.7)) that the Hausdorff distance between D ( t − ε ′ ) and D ( t )is at most ε ′ k ∗ max max θ | a ( θ ) | so that if ε ′ is chosen such that ε ′ k ∗ max max θ | a ( θ ) | < δ/ ε <ε ( δ/ , k ∗ max ). (cid:3) Upper bound: Proof of (6.3) .Definition 6.3. Define ( P i ( t )) i =1 to be the four “poles” of D ( t ) , where the tangent vectoris either horizontal or vertical (recall that D ( t ) is strictly convex at all times under our as-sumptions, cf. discussion after (5.12) , so that the four poles are distinct and uniquely defined). P ( t ) denotes the “north pole” and the others are numbered in the clockwise order. Denote by ( x ( P i ( t )) , y ( P i ( t ))) (resp. ( u ( P i ( t )) , v ( P i ( t ))) ) the coordinates of P i ( t ) in the coordinate system ( f , f ) (resp. ( e , e ) ). When t = 0 we omit the time coordinate. An equivalent formulation of (6.3) is: for all δ > ε small enough w.h.p. σ x ( εL ) = + for every x ∈ L [ D ( ε (1 − δ ))] c . (6.9)Given some small ξ we divide [ D ( ε (1 − δ ))] c in eight pieces ( M i ) i =1 and ( N i ) i =1 as follows (thisis analogous to the definition (4.16) in the scale-invariant case, cf. Figure 6): M ( ε, ξ ) := (cid:0) [ u ( P ) + ξ, ∞ ) × [ v ( P ) + ξ, ∞ ) (cid:1) \ D ( ε (1 − δ )) (6.10)while N ( ε, ξ ) is the infinite component of ([ u ( P ) − ξ, u ( P )+ ξ ] × R ) \D ( ε (1 − δ )) which contains P . The sets M i , N i are defined analogously for i = 2 , , 4, so that [ D ( ε (1 − δ ))] c = S i =1 ( M i ∪ N i ).Equation (6.9) is proved if one can prove that for every i , and ε small enough, w.h.p. σ x ( εL ) = + for every x ∈ LM i ( ε, ξ ) (6.11) σ x ( εL ) = + for every x ∈ LN i ( ε, ξ ) . (6.12)Of course one can focus on i = 1, the other cases being obtained by a permutation of coordinates.6.1.1. Proof of (6.11) . We use the notation f ( · , t ) for the function whose graph in the coordinatesystem ( f , f ) is the portion of ∂ D ( t ) which goes in the anti-clockwise direction from point A where the tangent forms an angle π/ B where the angle is (5 / π . The domain of definition of f ( · , t ) decreases with time (because D ( t )shrinks) but for t small enough it includes [ x ( P ) , x ( P )]. Let D be the “triangular-shaped”region delimited by ∂ D , by the vertical line ℓ passing through P and by the horizontal line ℓ passing through P (note that D may not be included in D ).We consider a modified dynamics in the north-east quadrant [ Lu ( P ) , ∞ ) × [ Lv ( P ) , ∞ ) de-limited by the lines Lℓ , Lℓ . All the spins are initially “ − ” in L D and “+” otherwise. As forboundary spins, the spins at distance at most 1 to the left of Lℓ are frozen to “ − ” if they arebelow LP and to “+” if they are above. The spins at distance at most 1 below Lℓ are frozento “ − ” if they are to the left of LP and to “+” otherwise, see Figure 12. In the quadrant underconsideration, this dynamics dominates the original one (for the inclusion order of the set of URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 35 +++++−−−−− − − − − − − − − PSfrag replacements Supressed column ℓ ℓ f f e e h ( y, t ) f ( x, t ) 2 ξ ξx AB yP P P P N ( ε, ξ ) M ( ε, ξ )0 Figure 12. The larger convex set is D and the smaller one is D ( ε (1 − δ )). Thepoles P i of D are marked with black dots (for convenience we have chosen P onethe vertical axis and P on the horizontal one). The graph in ( f , f ) of the anti-clockwise portion of ∂ D between A and B is f ( · , 0) and the graph in ( e , e ) ofthe portion of ∂ D between P and P is h ( · , ℓ are set to “ − ” below P and “+” above; boundary spinsbelow ℓ are set to “ − ” to the left of P and “+” to the right.“ − ” spins). Let F L ( · , t ) denote the function whose graph in ( f , f ) is the interface between “ − ”and “+” spins for this dynamics. Using exactly the same argument as in (4.22) we get thatlim L →∞ sup x ∈ [ x ( P ) ,x ( P )] sup t T (cid:12)(cid:12)(cid:12)(cid:12) L F L ( xL, tL ) − g ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 (6.13)in probability, where g is the solution for t > x ∈ ( x ( P ) , x ( P )) of ∂ t g ( x, t ) = ∂ x g ( x, t ) g ( · , t ) = f ( · , g ( x ( P ) , t ) = y ( P ) and g ( x ( P ) , t ) = y ( P ) . (6.14)We are thus reduced to prove that for every ˜ x , ˜ x satisfying x ( P ) < ˜ x < ˜ x < x ( P ) andevery x ∈ ( ˜ x , ˜ x ) g ( x, ε ) < f ( x, (1 − δ ) ε ) . (6.15) Lemma 4.3 (which is valid also in this case, since the curvature is Lipschitz and therefore ∂ x f ( · , η , for ε small enough, g ( x, ε ) f ( x, 0) + ε (cid:0) ∂ x f ( x, 0) + η (cid:1) . (6.16)We are left to estimate the right-hand side of (6.15). For any θ ∈ (0 , π/ 2) and s > x ( θ, s ) to be the f coordinate, in the ( f , f ) coordinate system, of the point of γ ( s ) where theoutward normal vector forms an anticlockwise angle θ with the horizontal vector e . Note thatfor s > x ( · , s ) defines a bijective function. We denote θ ( · , s ) its inverse.It is more practical for the purposes of this section to rewrite the curve-shortening flow inthe ( f , f ) coordinate system. Using the explicit expression (2.4) of a ( θ ), some trigonometryand the expression | f ′′ ( x ) | / (1 + ( f ′ ( x )) ) / for the absolute value of the curvature at the point( x, f ( x )) of the curve given by the graph of a function x f ( x ), one gets that for θ ∈ (0 , π/ a ( θ ) k ( θ, s ) = − ∂ x f ( x ( θ, s ) , s ) cos( θ − π ∂ t f ( x, s ) = − a ( θ ( x, s )) k ( θ ( x, s ) , s )cos( θ ( x, s ) − π ) = 14 ∂ x f ( x, s ) , (6.18)so that f ( x, (1 − δ ) ε ) = f ( x, 0) + Z (1 − δ ) ε ∂ x f ( x, s )d s. (6.19)We need therefore to prove time-regularity of ∂ x f ( · , s ): Lemma 6.4. One has sup {| ∂ t f ( x, s ) − ∂ t f ( x, | , s ∈ [0 , t ] and x ∈ [ x ( P ) , x ( P )] } ≤ Ψ( t, k max , k min , L ( k )) (6.20) where Ψ tends to zero with the first argument.Proof of Lemma 6.4. Recall from Section 5 that the curvature function k ( θ, s ) is jointly contin-uous in ( θ, s ) and that its modulus of continuity depends only on k max , k min , L ( k ). Thus usingequation (6.18) it is sufficient to prove that θ ( x, s ) is a continuous function in s uniformly in x :sup {| θ ( x, s ) − θ ( x, | , s ∈ [0 , t ] and x ∈ [ x ( P ) , x ( P )] } ≤ Ψ ( t, k max , k min , L ( k )) (6.21)where again Ψ tends to zero as t → 0. This comes from the continuity of x ( θ, · ):sup {| x ( θ, s ) − x ( θ, | , s ∈ [0 , t ] and θ ∈ [0 , π } ≤ Ψ ( t, k max , k min , L ( k )) (6.22)and from the fact that x ( · , s ) is strictly monotone: for t > {| ∂ θ x ( θ, s ) | , s t, θ ∈ [0 , π } > c ( k min ) > . (6.23)Both properties are a consequence of x ( θ, t ) = x ( π/ , t ) − Z θπ/ cos( θ ′ − π/ θ ′ k ( θ ′ , t ) (6.24)which is easily derived from (5.3)-(5.4). (cid:3) We finally get that for x ∈ ( x ( P ) , x ( P )) and ε small enough (as a function of k min , k max , L ( k )), f ( x, (1 − δ ) ε ) ≥ f ( x, 0) + (1 − δ ) ε ∂ x f ( x, − η ) . (6.25)Thus, combining this with (6.16), (6.15) is proved if one has ∂ x f ( x, 0) + η < (1 − δ ) (cid:0) ∂ x f ( x, − η (cid:1) (6.26) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 37 i.e. 2 η + δ∂ x f ( x, . (6.27)For this it is sufficient to have η small enough, since (cf. (6.18)) sup { ∂ x f ( x, , x ∈ [ x ( P ) , x ( P )] } can be upper bounded by a negative constant times the minimal curvature k min , which is strictlypositive. (cid:3) Proof of (6.12) . Set h ( · , t ) to be the continuous concave function whose graph in the( e , e ) coordinate system is the portion of γ ( t ) which goes from P ( t ) to P ( t ) with the anti-clockwise orientation. Given a small η choose ξ small enough so that sup {| ∂ x h ( x, | , u ( P ) − ξ x u ( P ) + ξ } ≤ η. Consider ¯ h ( · ) the C function equal to h ( · , 0) on [ u ( P ) − ξ, u ( P ) + 2 ξ ] and affine outside.Assume for definiteness that ¯ h ( u ( P ) − ξ ) ≤ ¯ h ( u ( P ) + 4 ξ ). Define ξ − = u ( P ) − ξ and ξ + = inf { x > u ( P ) , ¯ h ( x ) = ¯ h ( ξ − ) } . We consider the restriction of ¯ h to [ ξ − , ξ + ] and still call it¯ h . Define J := [ ξ + , ∞ ) × [¯ h ( ξ + ) , ∞ ) , J := ( −∞ , ξ − ] × [¯ h ( ξ + ) , ∞ ) . (6.28)We consider the same chain of monotonicities as in the scale-invariant case (Section 4.3.2) andwe end up with a dynamics in the half-strip [ Lξ − , Lξ + ] × [ L ¯ h ( ξ + ) , ∞ ) with boundary spins frozento “+” in L ( J ∪ J ) and to “ − ” in Z ∗ × ( −∞ , L ¯ h ( ξ + )] and an initial condition with “ − ” spinsunder the graph of L ¯ h ( · /L ). Also, the dynamics thus obtained does not allow moves that makethe interface non-connected. Calling ( σ ( t )) t > this dynamics, (4.35) is satisfied.Define H L : [ Lξ − , Lξ + ] → Z to be the function whose graph in ( e , e ) is the interface between“+” and “ − ” spins. We have to prove1 L H L ( Lx, εL ) ≤ h ( x, (1 − δ ) ε ) for every x ∈ ( u ( P ) − ξ, u ( P ) + ξ ) . (6.29)Following the same steps as in (4.37) to (4.40) (recall that ∂ x h ( · , 0) is uniformly continuous bythe Lipschitz curvature assumption) one finds that the left-hand side of (6.29) is upper boundedw.h.p. by h ( x, 0) + ε η ) − ( ∂ x h ( u ( P ) , 0) + r ( x, L ( k ))) + o ( ε ) , (6.30)where r ( x, L ( k )) tends to 0 when x → u ( P ).To estimate the r.h.s of (6.29), one remarks that, in analogy with (6.18), ∂ s h ( x, s ) = − a ( θ ( x, s )) k ( θ ( x, s ) , s ) / sin( θ ( x, s )) (6.31)so that ∂ t h ( x, t ) is continuous in x and t (since θ is around π/ 2, sin( θ ( x, s )) is bounded awayfrom zero). Moreover ∂ t h ( u ( P ) , 0) = 12 ∂ x h ( u ( P ) , , (6.32)which can be obtained directly from a (0) = 1 / D atthe north pole P equals minus the second derivative of h ( x, 0) computed at x = u ( P ). Thusfor every x ∈ ( u ( P ) − ξ, u ( P ) + ξ ) h ( x, (1 − δ ) ε ) > h ( x, 0) + (1 − δ ) ε η ) ∂ x h ( u ( P ) , , (6.33)and (6.29) is proven (combining (6.30) and (6.33)) choosing η and ξ small enough. Lower bound: Proof of (6.4) . We are confident that the reader is by now convincedthat the proof of Theorem 2.2 is essentially identical to that in the scale-invariant case, modulothe fact that the definitions of the various subsets of R needed to define the regions where spinsare frozen to “ − ” or “+” ( U, J , J , etc) have to be adapted in the obvious way due to the lackof discrete-rotation symmetry of the general initial droplet D . We will therefore skip altogetherthe proof of (6.4) and we limit ourselves to indicating the only point where some (minor) carehas to be taken.The definition (4.45) of the set U is replaced by U := D ( − ν ) , cf. (2.8). Let s be the verticalsegment obtained moving downwards from the “north pole of U ” until the point c where s meets s , the horizontal segment obtained moving to the left from the “east pole” of U until c is reached. To prove the analog of (4.48), mimicking the proof given in Section 4.4.4, onewould like to apply (6.2) in order to freeze to “ − ” all the spins along the two rescaled segments Ls , Ls . This is however not allowed in general, because nothing guarantees that they areentirely contained in LU , i.e., that c ∈ U (this problem does not occur for the invariant shape D , where c is the origin). The solution however is simple (cf. Figure 13): one just freezes to “ − ”all the spins along the portions of Ls , Ls which are inside LU , and along the shorter portionof L∂U which connects them (call Γ this portion). The point is that in this situation the + / − interface between north and east poles follows again the corner dynamics and Theorem 3.2 isapplicable. The freezing of “ − ” spins along Γ is equivalent to putting a hard-wall constraint inthe corner dynamics (the interface is not allowed to cross a zig-zag path which approximates Γ)but this is irrelevant: since Γ is at distance of order L away from the linear profile the cornerdynamics approaches for long times, the probability that the interface even feels the hard-wallconstraint within the diffusive times of order L we are interested in goes to zero with L (thisagain can be seen via Theorem 3.2). Other than that, the proof of (6.4) is identical to that inthe D = D case. PSfrag replacements Ls Ls c LU = L D ( − ν ) Ls Γ L D Figure 13. URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 39 Proof of Theorem 3.2: scaling limit for SSEP The first step is to discretize (3.6), so that instead of working with φ ( · , · ) we get Φ( · , · ) solutionof the analogous discrete Cauchy problem: ∂ t Φ( x, t ) = ∆Φ( x, t )Φ L (0 , t ) = h = 0Φ L ( L, t ) = h L Φ L ( x, 0) = h x (7.1)for every t ≥ x ∈ { , . . . , L − } . Here ∆ is the discrete Laplacian operator:(∆ f )( x ) := f ( x + 1) + f ( x − − f ( x ) ∀ x ∈ { , . . . , L − } . (7.2)Note that Φ( x, t ) = E [ h x ( t )], with ( h ( t )) t ≥ the process with generator (3.3), and that Φ(0 , t ) − h ( t ) = Φ( L, t ) − h L ( t ) = 0. It is a standard result that Φ, solution of the discrete space heat-equation, converges to φ in all reasonable norms when L → ∞ in the diffusive limit. We recordthis result here: Lemma 7.1. lim L →∞ max t ∈ [0 ,T ] max x ∈ [0 , L | Φ( ⌊ xL ⌋ , tL ) − Lφ ( x, t ) | = 0 . (7.3)Using Lemma 7.1, we are reduced to provelim L →∞ P (cid:20) max t ∈ [0 ,T L ] max x ∈{ ,...,L − } | h x ( t ) − Φ( x, t ) | < εL (cid:21) = 1 . (7.4)Both h · ( t ) and Φ( · , t ) are 1-Lipschitz functions (for all t ) so that | h · ( t ) − Φ( · , t ) | is 2-Lipschitzand (cid:26) max x ∈{ ,...,L − } | h x ( t ) − Φ( x, t ) | > a (cid:27) = ⇒ ( L − X x =1 [ h x ( t ) − Φ( x, t )] > a / ) . (7.5)As a consequence, (7.4) is equivalent to prove the following L convergence statement: Proposition 7.2. The following convergence in probability holds: lim L →∞ sup t ∈ [0 ,L T ] L L − X x =1 [ h x ( t ) − Φ( x, t )] = 0 . (7.6) Proof of Proposition 7.2. The restriction of the operator ∆ toΛ L = { g : { , . . . , L } 7→ R , g (0) = g ( L ) = 0 } is self-adjoint (for the canonical scalar product on R L − denoted in the sequel by h· , ·i ) and thefamily of functions f k : { , . . . , L } ∋ x r L sin (cid:18) kπxL (cid:19) , k = 1 , . . . , L − L of ∆-eigenfunctions, with respective eigenvalues − λ k := 2 cos (cid:18) πkL (cid:19) − < . (7.8) As the function x h x ( t ) − Φ( x, t ) is in Λ L it can be decomposed on this basis. We use thenotation H kt for its k − th coordinate (multiplied by p L/ H kt := L X x =0 [ h x ( t ) − Φ( x, t )] sin (cid:18) kπxL (cid:19) . (7.9)The quantity one wants to estimate in (7.6) is equal tosup t ∈ [0 ,L T ] L L − X k =1 ( H kt ) ≤ L L − X k =1 sup t ∈ [0 ,L T ] ( H kt ) . (7.10)We control the right-hand side by controlling each H kt separately. Lemma 7.3. For every L , k ∈ { , . . . , L − } and t > one has deterministically | H kt | ≤ L /k. (7.11) Moreover for any given T , w.h.p. | H kt | ≤ L / for every k ≤ (log L ) / and every t ≤ L T. (7.12) Proof of Lemma 7.3. The first point is easy. Using summation by parts H kt = L X x =1 ([ h x ( t ) − Φ( x, t )] − [ h x − ( t ) − Φ( x − , t )]) L X y = x sin (cid:18) kπyL (cid:19) . (7.13)Then one can check that for every x and k | [ h x ( t ) − Φ( x, t )] − [ h x − ( t ) − Φ( x − , t )] | ≤ , (7.14) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X y = x sin (cid:18) kπyL (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Lk (7.15)so that (7.11) follows.For the second point, first, one notices that for all k ∈ { , . . . , L − } , the functions F k : Ω M L ,N L ∋ h L X x =0 sin (cid:18) πkxL (cid:19) (cid:20) h x − h L − h L x (cid:21) (7.16)are eigenfunctions of L with respective eigenvalues − λ k / 2. Indeed F k is just a linear combinationof the coordinate function A x : h h x (plus a constant), and it can be seen from the verydefinition (3.3) of the generator L , that L ( A x )( h ) = 12 (∆ h )( x ) = 12 (∆˜ h )( x ) (7.17)using the notation ˜ h x = h x − h L − h L x . Hence (note that ˜ h ∈ Λ L )2 L F k ( h ) = p L/ h f k , ∆˜ h i = p L/ h ∆ f k , ˜ h i = − λ k p L/ h f k , ˜ h i = − λ k F k ( h ) . (7.18)As a consequence one can rewrite H kt = L X x =0 sin (cid:18) kπxL (cid:19) ˜ h x ( t ) − e − λ k t/ L X x =0 sin (cid:18) kπxL (cid:19) ˜ h x (0) (7.19)and notice that M kt := e λ k t/ H kt is a martingale. Therefore one can get the result by computingthe second moment of M kt and using Doob’s inequality. URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 41 It is not difficult to bound the quadratic variation of M k . Notice that E h ( M kt ) i = E (cid:20)Z t d h M k i s (cid:21) (7.20)and thatd h M k i s = e λ k s d h H k i s = e λ k s L − X k =1 sin (cid:18) kπxL (cid:19) (∆( h ( t ))( x )) s ≤ Le λ k s d s (7.21)so that E (cid:2) ( M kt ) (cid:3) ≤ L R t e λ k s d s. Therefore (using λ k = π k /L (1 + o (1)) uniformly for all k (log L ) / ), P " sup t ∈ [0 ,L T ] | H kt | ≥ a ≤ P " sup t ∈ [0 ,L T ] | M kt | ≥ a ≤ C L e λ k L T a k . (7.22)Using this inequality for a := L / and all k (log L ) / one gets that P h ∃ t ∈ [0 , L T ] , ∃ k (log L ) / , | H kt | ≥ L / i ≤ X k (log L ) / Ck √ L e k π T . (7.23)One can check that the right-hand side above tends to zero when L goes to infinity, whichfinishes the proof of Lemma 7.3. (cid:3) We now turn to (7.10):2 L L − X k =1 sup t ∈ [0 ,L T ] ( H kt ) ≤ L X k (log L ) / sup t ∈ [0 ,L T ] ( H kt ) + 32 L X k = ⌈ (log L ) / ⌉ k − . (7.24)The second term tends to zero (it is roughly (log L ) − / ). The first one is w.h.p. less than2 L X k (log L ) / L / log L √ L . (7.25)This achieves the proof of Proposition 7.2 and thus also the one of Theorem 3.2. (cid:3) Appendix A. Proof of Theorem 3.4: scaling limit for the zero-range process This section follows quite closely computations in Appendix A of [23].A.1. Particle system and monotonicity. For x ∈ {− L, . . . , L } we denote η x := h x +1 − h x the discrete gradient of h in x . A configuration h ∈ Ω L can be alternatively given by η ∈ Θ L := { η : {− L, . . . , L } → Z } . It turns out that the z ero-range process description of the dynamics(cf. Section 3.2) is easier to work with.For a more formal description of the dynamics we write explicitly its generator. For η ∈ Θ L and x ∈ {− L, . . . , L − } , we define the configuration → η ( x ) as → η ( x ) ( x ) := η x − sg( η x ) , → η ( x ) ( x + 1) := η x +1 + sg( η x ) , → η ( x ) ( y ) := η y , ∀ y / ∈ { x, x + 1 } . (A.1) We define ← η ( x ) analogously for x ∈ {− L + 1 , . . . , L } replacing x + 1 in the second and third linesby x − 1. The sign function sg is given bysg( a ) := a > , − a < , a = 0 . (A.2)The generator of the chain seen in the state-space Θ L is given by L f := 12 L − X x = − L (cid:20) f ( → η ( x ) ) + f ( ← η ( x +1) ) − f ( η ) (cid:21) . (A.3)Note that the dynamics conserves the sum of the η ’s, i.e. the value of h L +1 .Before going to the core of the proof, we need to change slightly the initial condition. In orderto compare with the original one, one needs the following monotonicity statement: Proposition A.1. [Coupling] (i) There is a canonical way of constructing simultaneously the dynamics with generator (3.8) from all possible initial configurations h . It satisfies the following monotonicityproperty: given h and ¯ h with h x ≥ ¯ h x for all x , the dynamics h and ¯ h starting from h and ¯ h respectively satisfy h x ( t ) ≥ ¯ h x ( t ) for every t and x . Moreover, the dynamicsstarted from h + a , a ∈ Z , (a vertically translated version of h , including the boundaryconditions h and h L +1 ), is simply ( h ( t ) + a ) t ≥ . (ii) There is a canonical way of constructing the dynamics with generator (A.3) from allpossible initial configurations η . It satisfies the following monotonicity property: given η and ¯ η with η x ≥ ¯ η x for all x , the dynamics η and ¯ η starting from η and ¯ η respectivelysatisfy η x ( t ) ≥ ¯ η x ( t ) for every t and x .Proof. The idea of the proof is using a canonical construction of the process, similarly to whatis done in Section 2.3. It is quite classic but we perfom it here for the sake of completeness. • For x ∈ {− L + 1 , L } we define ( τ n,x ) n ≥ and ( τ ′ n,x ) n ≥ to be two IID clock processes,with τ ,x = 0 and τ n +1 ,x − τ n,x IID exponential variables of mean 2. • The process h ( · ) is c`adl`ag and constant in time except at the of the ringing times ofthe clock processes. At time τ n,x only h x is modified, as follows: h x ( τ n,x ) = h x ( τ − n,x ) +sg( h x − ( τ − n,x ) − h x ( τ − n,x )), the other coordinates being left unchanged. At time τ ′ n,x only h x is modified, as follows: h x ( τ ′ n,x ) = h x (( τ ′ n,x ) − ) + sg( h x +1 (( τ ′ n,x ) − ) − h x (( τ ′ n,x ) − )), theother coordinates being left unchanged.The reader can check that this allows to couple the dynamics from all possible initial conditionsand that our coupling has the desired properties. This coupling induces a coupling on η thatalso has the right properties. (cid:3) A.2. Changing the initial condition. We prove (3.14) working with an initial conditionwhich is not the one, h , described in (3.11), which is random and for which the number ofparticle at a site is given by a geometric variable. The reason for this change of initial conditionwill appear in the proof of ( iii ) in Lemma A.3. We explain in this section why this implies theresult starting from h .Given a continuous function φ : [ − , → R with φ ( ± 1) = 0 and with a finite number ofchanges of monotonicity, set (ˆ η x ) x ∈{− L,...,L } to be a family of independent variables with thefollowing distribution: if φ (( x + 1) /L ) − φ ( x/L ) ≥ η x is a geometric variable of mean URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 43 L ( φ (( x +1) /L ) − φ ( x/L )) and if ( φ (( x +1) /L ) − φ ( x/L )) < − ˆ η x is a geometric variableof mean L ( φ ( x/L ) − φ (( x + 1) /L )) (with the convention that φ (1 + 1 /L ) = 0). One setsˆ h x = x − X y = − L ˆ η y . (A.4)Note that for every ε > 0, w.h.p,ˆ h x − L / ε ≤ h x ≤ ˆ h x + L / ε for every x ∈ {− L, . . . , L + 1 } . (A.5)Let ( h ( t )) t ≥ , (ˆ h ( t )) t ≥ be the dynamics with generator (3.8) started with initial condition h ,ˆ h respectively, constructed using the canonical way of Proposition A.1 (i). Then with highprobability, for every t > x ∈ {− L, . . . , L } ˆ h x ( t ) − L / ε ≤ h x ( t ) ≤ ˆ h x ( t ) + L / ε . (A.6)Therefore in order to prove (3.14) for h ( · ), it is sufficient to prove it for ˆ h ( · ). We let ˆ η x ( t ) =ˆ h x +1 ( t ) − ˆ h x ( t ) denote the gradient of ˆ h .A.3. Proof of an L statement. For (ˆ h ( t )) t ≥ defined above one has Proposition A.2. For any t ≥ L →∞ E " L L +1 X x = − L (Φ( x, L t ) − ˆ h x ( L t )) = 0 (A.7)This result does not directly imply (3.14) (ˆ h may have a priori unbounded gradients), but itis not to difficult conclude from Proposition A.2, see Section A.4. In the rest of the section, forlightness of notation we write h, η instead of ˆ h, ˆ η .Before starting the proof we need some technical statements. First note, recalling the defini-tion of the generator (3.8), that for every x ∈ {− L + 1 , . . . , L } ∂ t E [ h x ( t )] = E [sg( η x ( t )) − sg( η x − ( t ))] , ∂ t E (cid:2) h x ( t ) (cid:3) = E [2 h x ( t )(sg( η x ( t )) − sg( η x − ( t ))) + ( | sg( η x ( t )) | + | sg( η x − ( t )) | )] . (A.8)Now some remarks: Lemma A.3. The following properties hold (recall notations in (3.13) ): (i) max x | q x ( t ) | is a non-increasing function of t . As a consequence ∀ t > , ∀ x ∈ {− L, . . . , L } , | q x ( t ) | ≤ k ∂ x φ k ∞ . (A.9)(ii) max x | σ ( q x +1 ( t )) − σ ( q x ( t )) | is a non-increasing function of t (recall σ ( u ) = u/ (1 + | u | ) ).Then, using also (i), for some C ( φ ) = C ( k ∂ x φ k ∞ , k ∂ x φ k ∞ ) < ∞ one has ∀ t > , ∀ x ∈ {− L, . . . , L } , | q x +1 ( t ) − q x ( t ) | ≤ C ( φ ) /L. (A.10)(iii) For any t , the random vectors ( η x ( t )) x ∈{− L,...,L } and ( − η x ( t )) x ∈{− L,...,L } are stochasticallydominated by L + 1 IID geometric variables with mean k ∂ x φ k ∞ .Proof. For ( i ) it is sufficient to show that Q ( t ) = max x q x ( t ) is non-increasing (by a similarargument one shows that min q x ( t ) is non-decreasing). As the maximum over finitely manydifferentiable functions, max x q x ( t ) possesses a right and a left-derivative everywhere and theright-derivative is equal to ∂ + t Q ( t ) = max x ∈ argmax q · ( t ) ∂ t q x ( t ) . (A.11) For any x in max x ∈ argmax q · ( t ) , one has2 ∂ t q x ( t ) = σ ( q x +1 ( t )) + σ ( q x − ( t )) − σ ( q x ( t )) ≤ , (A.12)(as σ ( q x ( t )) is maximal), and therefore Q ( t ) is decreasing.For ( ii ): Using the same argument as for the point ( i ), we have to note that for any fixedtime T and x where max x [ σ ( q x +1 ) − σ ( q x )]( T ) is attained one has2 [ ∂ t { σ ( q x +1 ) − σ ( q x ) } ] ( T ) = σ ′ ( q x +1 ( T )) [ σ ( q x +2 ( T )) − σ ( q x +1 ( T ))]+ σ ′ ( q x ( T )) [ σ ( q x ( T )) − σ ( q x − ( T ))] − (cid:2) σ ′ ( q x +1 ( T )) + σ ′ ( q x ( T )) (cid:3) [ σ ( q x +1 ( T )) − σ ( q x ( T ))] ≤ . (A.13)Therefore, one has that | σ ( q x +1 ( t )) − σ ( q x ( t )) | ≤ C ( φ ) L . In order to deduce (A.10), write σ ( q x +1 ( t )) − σ ( q x ( t )) = σ ′ ( y ) [ q x +1 ( t ) − q x ( t )]for some q x +1 ( t ) ≤ y ≤ q x ( t ). Since the q x are bounded (point ( i )) and σ ( · ) has uniformlypositive derivative on bounded intervals, (A.10) follows.For ( iii ): One has that L (cid:0) φ ( x +1 L ) − φ ( xL ) (cid:1) ≤ k ∂ x φ k ∞ so that the initial configuration η isstochastically dominated by ˜ η the configuration given by 2 L + 1 IID geometric variables withmean k ∂ x φ k ∞ . According to Proposition A.1 (ii), one can couple the two dynamics η and ˜ η starting from η and ˜ η so that η ( t ) ≤ ˜ η ( t ) for all t ≥ 0. For fixed t the law of ˜ η ( t ) is the sameas the one of ˜ η as this distribution is stationary for the dynamics. The other domination isproved in the same way. (cid:3) Proof of Proposition A.2. We estimate the difference between E h L P L +1 x = − L (Φ( x, L t ) − h x ( L t )) i and the same quantity at time zero, by considering it as the integral of its time-derivative. URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 45 E " L L +1 X x = − L (Φ( x, L t ) − h x ( L t )) − E " L L +1 X x = − L (Φ( x, − h x (0)) = 1 L Z L t L X x = − L +1 ∂ s E (cid:2) (Φ( x, s ) − h x ( s )) (cid:3) d s = 1 L L X x = − L +1 Z L t E { (Φ( x, s ) − h x ( s )) ( σ ( q x ( s )) − σ ( q x − ( s ))) − Φ( x, s )(sg( η x ( s )) − sg( η x − ( s )))+ h x ( s )(sg( η x ( s )) − sg( η x − ( s ))) + 12 ( | sg( η x ( s )) | + | sg( η x − ( s ) | ) (cid:27) d s = 1 L L X x = − L Z L t E h − q x ( s ) σ ( q x ( s )) + η x ( s ) σ ( q x ( s )) + q x ( s )sg( η x ( s )) − ( | η x ( s ) | − | sg( η x ( s ) | ) i d s − L Z L t E (cid:20) h L +1 ( s )(sg( η L ( s )) − σ ( q L ( s ))) + 12 ( | sg( η − L ( s ) | + | sg( η L ( s ) | ) (cid:21) d s. (A.14)The second equality is obtained by expanding the product and using (A.8) and (3.12) to estimateall the derivated terms. The third equality is obtained via summation by parts, it gives a termthat is due to boundary effect (the second one) which can be bounded as follows L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L t E (cid:20) h L +1 ( s )(sg( η L ( s )) − σ ( q L ( s ))) + 12 ( | sg( η − L ( s )) | + | sg( η L ( s )) | ) (cid:21) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CL − (1 + E | h L +1 | ) = O ( L − / ) . (A.15)Indeed h L +1 ( t ) is constant through time and is the sum of 2 L + 1 independent variables. Themean of this sum is 0 and the variance of each term is bounded as we supposed φ to be smooth.The variance of h L +1 is thus O ( L ). We can also neglect the second term in the first line as E " L L +1 X x = − L (Φ( x, − h x (0)) = 1 L L +1 X x = − L Var( h x (0)) = O ( L − ) (A.16)where the last equality is easy to obtain once noticed that h x is the sum of ( L + x ) independentgeometric variables with bounded variance.Set A ( x, s ) := − q x ( s ) σ ( q x ( s )) + η x ( s ) σ ( q x ( s )) + q x ( s )sg( η x ( s )) − ( | η x ( s ) | − | sg( η x ( s ) | ) . (A.17)From the previous equations one gets that E " L L X x = − L +1 (Φ( x, L t ) − h x ( L t )) = 1 L Z L t L X x = − L +1 E [ A ( s, x )]d s + o (1) . (A.18)To understand better the rest of the proof, the reader should notice that if ( η x ( s )) x ∈{− L,...,L } were distributed like geometric variables it would be possible to factorize E [ A ( x, s )] in a product of negative sign and from equation (A.18) the proof would be over. Indeed, for q > η distributed like a geometric variable of mean u > − η is distributed like a geometric variableof mean − u > E [ − qσ ( q ) + ησ ( q ) + q sg( η ) − ( | η | − | sg( η ) | )] = − ( q − u )( σ ( q ) − σ ( u )) ≤ , (A.19)(recall that σ ( · ) is an increasing function). It is not true in general that η x ( s ) are geometricallydistributed for s > η x ( s ) (using this space-time average is somehow crucial for the proof to work).As the limiting object is an infinite volume measure, it is somehow more convenient to consider η ( s ) as an element of Z Z by periodizing it: for the system of size (2 L +1) one sets η x + k (2 L +1) = η x for every k ∈ Z , x ∈ {− L, . . . , L } . For y ∈ Z one defines θ y to be the shift operator η θ x η defined by ∀ x ∈ Z , ( θ y η ) x := η x + y . (A.20)We define for each L > µ Lt on Z Z our space-time averaged measures by itsaction on local functions (for K ∈ N we call f ( η ) a K -local function if f is bounded and can bewritten as a function of η | [ − K,K ] ; f is a local function if there exists a K such that it is K -local): µ Lt ( f ) := E tL L + 1 Z L t L X y = − L f ( θ y ( η ( s )))d s , (A.21)We want to prove that any limit point (when L → ∞ ) of µ Lt is an equilibrium and use thisinformation to bound the right-hand side of (A.18).We introduce some notation to describe the limiting measure. For u ∈ R define ρ u to be ameasure on η = ( η x ) x ∈ Z such that the η x are IID geometric variables of mean u if u ≥ − η x are IID geometric variables of mean − u if u < 0. If ν is a probability measure on R define ρ ν := Z ρ u ν (d u ) . (A.22) Proposition A.4. Fix t > . For any subsequence of ( µ L n t ) n ≥ , it is possible to find a sub-subsequence ( µ L ′ n t ) n ≥ that converges locally to ρ ν with ν a probability measure on R with supportincluded in [ −k ∂ x φ k ∞ , k ∂ x φ k ∞ ] , in the sense that for any local function f lim n →∞ µ L ′ n t ( f ) = ρ ν ( f ) . (A.23) As a consequence for any local function f lim sup L →∞ µ Lt ( f ) ≤ max u ∈ [ −k ∂ x φ k ∞ , k ∂ x φ k ∞ ] ρ u ( f ) . (A.24) Remark A.5. Note that the convergence does not hold in the topology induced by the totalvariation distance: indeed µ Lt give mass one to L -periodic η whereas these configurations havemass zero for the limiting measure. Proof of Proposition A.4. For any fixed K > 0, the sequence of laws of ( η x ) x ∈ [ − K,K ] under µ L n t is tight by Lemma A.3 ( iii ) and hence we can extract a converging subsequence. By diagonal URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 47 extraction it is possible to extract a subsequence L ′ n of L n and a family of measures ( µ K ) K > on Z [ − K,K ] such that the law of ( η x ) x ∈ [ − K,K ] under µ L ′ n t converges to µ K for all K . By constructionfor H > K , µ H projected on Z [ − K,K ] is equal to µ K and by Kolmogorov extension theorem,there exists a measure µ on Z Z such that µ projected on Z [ − K,K ] equals µ K for all K . One hastherefore for all local function f lim n →∞ µ L ′ n t ( f ) = µ ( f ) . (A.25)We have to show that µ can be written as ρ ν . First one remarks that µ L n t is translation invariant,so that µ is too. A second point to make is that µ -almost surely all the η x (that are not equalto zero) have the same sign. Indeed µ (cid:0) ∃ x, x ′ ∈ Z , η x η ′ x < (cid:1) = lim K →∞ µ (cid:0) ∃ x, x ′ ∈ [ − K, K ] , η x η ′ x < (cid:1) = lim K →∞ lim n →∞ µ L ′ n t (cid:0) ∃ x, x ′ ∈ [ − K, K ] , η x η ′ x < (cid:1) (A.26)and µ Lt (cid:0) ∃ x, x ′ ∈ [ − K, K ] , η x η ′ x < (cid:1) = 1 tL (2 L + 1) Z L t L X y = − L P (cid:2) ∃ x, x ′ ∈ [ − K + y, K + y ] , η x ( s ) η x ′ ( s ) < (cid:3) d s. (A.27)One realizes easily that L X y = − L {∃ x,x ′ ∈ [ − K + y,K + y ] ,η x η x ′ < } (A.28)is upper bounded by (2 K + 1) times the number of changes of sign in ( η x ) x ∈ [ − L,L +1] . Fromthe definition of the dynamics, a transition can only lower the number of changes of sign. Itsinitial value is smaller than the number of changes of monotonicity of φ (which is assumed tobe finite) plus one (the “plus one” can come from periodizing). Therefore L X y = − L P (cid:2) ∃ x, x ′ ∈ [ − K + y, K + y ] , η x ( s ) η x ′ ( s ) < (cid:3) ≤ KC ( φ ) . (A.29)A third point is to show is that µ is an invariant measure for the infinite volume dynamics(the infinite volume version of (A.3), call its generator L ∞ ). For f a K -local function one has(for L ≥ K large enough) µ Lt ( L ∞ f ) = 1 tL L + 1 Z L t L X y = − L E ( L ∞ ( f ◦ θ y )( η ( s ))) d s. (A.30)For y ∈ [ − L + K, L − K ] the infinite volume generator applied to f has the same effect as thefinite volume generator so that Z t E [ L ∞ ( f ◦ θ y )( η ( s ))] d s = Z t ∂ s E [( f ◦ θ y )( η ( s ))] d s = E [( f ◦ θ y )( η ( t )) − ( f ◦ θ y )( η (0))] . (A.31) Therefore µ Lt ( L ∞ f ) = 1 tL L + 1 L − K X y = − L + K E (cid:2) ( f ◦ θ y )( η ( tL )) − ( f ◦ θ y )( η (0)) (cid:3) + 1 tL L + 1 Z L t − L + K − X y = − L + L X y = L − K +1 E ( L ∞ ( f ◦ θ y )( η ( s ))) d s = O (1 /L ) . (A.32)As a consequence, for any local function µ ( L ∞ f ) = lim n →∞ µ L ′ n t ( L ∞ f ) = 0 . (A.33)Restricted on the event η x have all the same sign, L ∞ is the generator of the zero-range processwith one type of particle and therefore µ is a translation invariant measure for the zero-rangeprocess. From [1, Theorem 1.9] one can write µ = ρ ν for some ν . By Lemma A.3 ( iii ), under µ ,at time zero η is dominated by a IID family of geometric variables of mean k ∂ x φ k ∞ and so is − η . This implies the claim on the support of ν .The second point of Proposition A.4 is standard; we include its proof for completeness. Givena local f one can extract a subsequence L n such thatlim n →∞ µ L n t ( f ) = lim sup L →∞ µ Lt ( f ) . (A.34)From L n one can extract a subsequence L ′ n such that µ L ′ n t converges to ρ ν so thatlim n →∞ µ L n t ( f ) = lim n →∞ µ L ′ n t ( f ) = Z ρ u ( f ) ν (d u ) , (A.35)which ends the proof. (cid:3) Fix l to be a large fixed integer. Set for y ∈ Z B y := { y, . . . , l + y } . (A.36)For notational convenience, similarly to η with (A.20), one considers now periodized version( q x ( s )) s ∈ Z of q ( s ) and ( A ( x, s )) x ∈ Z of A ( · , s ).Now, one uses Proposition A.4 to control each term in E P A ( s, x ). Lemma A.6. lim l →∞ lim sup L →∞ L E Z L t L X y = − L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y q x ( s )sg( η x ( s )) − q y ( s ) σ l X x ∈ B y η x ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (A.37) Proof of Lemma A.6. Fix l > 0. For L large enough, any all y ∈ {− L, . . . , L − l } , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y q x ( s )sg( η x ( s )) − q y ( s ) σ l X x ∈ B y η x ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | q y ( s ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y sg( η x ( s )) − σ l X x ∈ B y η x ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + max x ∈ B y | q x ( s ) − q y ( s ) | . (A.38) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 49 Moreover, uniformly in y ∈ {− L, . . . , L − l } , as a consequence of Lemma A.3 ( ii )max y ∈{− L,...,L − l } ,x ∈ B y ,s ≥ | q x ( s ) − q y ( s ) | = O ( l/L ) . (A.39)The contribution of y ∈ { L − l + 1 , L } to the sum under the integral in (A.37) is O ( l ). Thereforesumming over y ∈ {− L, . . . , L } , integrating over s and taking expectation one gets (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z tL E L X y = − L l X x ∈ B y q x ( s )sg( η x ( s )) − q y ( s ) σ l X x ∈ B y η x ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (max y,s | q y ( s ) | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z tL E L X y = − L l X x ∈ B y sg( η x ( s )) − σ l X x ∈ B y η x ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( lL )= (max y,s | q y ( s ) | ) tL (2 L + 1) µ Lt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B sg( η x ) − σ l X x ∈ B η x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( lL ) (A.40)where µ Lt is defined in (A.21). Therefore, the proof of our statement is finished provided oneproves lim l →∞ lim sup L →∞ µ Lt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B sg( η x ) − σ l X x ∈ B η x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (A.41)From Proposition A.4 one haslim sup L →∞ µ Lt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B sg( η x ) − σ l X x ∈ B η x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup ≤ u ≤k ∂ x φ k ∞ ρ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B sg( η x ) − σ l X x ∈ B η x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A.42)and one can check that the right-hand side term tends to zero when l tends to infinity: we notethat under ρ u , for every x one has ρ u (sg( η x )) = σ ( u ), and the law of large numbers tell us thatthe two terms l P x ∈ B y sg( η x ) and σ (cid:16) l P x ∈ B y η x (cid:17) have the same limit when l tends to infinity.However, because of the sup over u one needs more quantitative estimates than the law of largenumbers to conclude. For instance we can get them by the use of second moment method; weleave the details to the reader. (cid:3) Similarly to Lemma A.6 one shows that Lemma A.7. lim l →∞ lim sup L →∞ L Z L t L X y = − L E ( G ( η ( s ))) = lim l →∞ lim sup L →∞ t (2 L + 1) L µ Lt ( G ( η )) = 0 (A.43) where G ( η ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y | η x ( s ) | − | sg( η x ( s ) | − l X x ∈ B y η x ( s ) σ l X x ∈ B y η x ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.44) Proof. The proof is very similar to that of Lemma A.6, the only additional technical point beingthat the function G ( η ) is not bounded so that one cannot use directly Proposition A.4. Howeverstochastic domination given by Lemma A.3 ( iii ) allows us to get the same conclusion by con-sidering the function η min( G ( η ) , K ), and letting K tend to infinity afterwards. Altogetherone gets lim sup L →∞ µ Lt ( G ) ≤ sup ≤ u ≤k ∂ x φ k ∞ ρ u ( G ) . (A.45)We end the proof in the same way that for the previous Lemma remarking that ρ u ( | η x | − | sg( η x ) | ) = uσ ( u ) . (cid:3) Now we are ready to conclude: L X y = − L A ( y, s )= L X y = − L − q y ( s ) σ ( q y ( s )) + 1 l X x ∈ B y η x ( s ) σ ( q x ( s )) + q x ( s )sg( η x ( s )) − ( | η x ( s ) | − | sg( η x ( s ) | ) ≤ L X y = − L − q y ( s ) σ ( q y ( s ) + l X x ∈ B y η x ( s ) σ ( q y ( s )) + q y ( s ) σ l X x ∈ B y η x ( s ) − l X x ∈ B y η x ( s ) σ l X x ∈ B y η x ( s ) + R ( s, l, L )= R ( s, l, L ) − L X y = − L q y ( s ) − l X x ∈ B y η x ( s ) σ ( q y ( s )) − σ l X x ∈ B y η x ( s ) (A.46)where R ( s, l, L ) = − L X y = − L l X x ∈ B y η x ( s )( σ ( q y ( s )) − σ ( q x ( s ))) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y q x ( s )sg( η x ( s )) − q y ( s ) σ l X x ∈ B y η x ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X x ∈ B y | η x ( s ) | − | sg( η x ( s ) | − l X x ∈ B y η x ( s ) σ l X x ∈ B y η x ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A.47)and the second term is non positive ( a − b and σ ( a ) − σ ( b ) have the same sign).According to ( ii ) − ( iii ) in Lemma A.3 (to control the first term) and Lemmata A.6 and A.7lim l →∞ lim sup L → L Z L t E R ( s, l, L )d s = 0 . (A.48) URVE-SHORTENING EVOLUTION FOR THE 2D ISING MODEL 51 This implies lim sup L →∞ L Z L t L X y = − L E A ( x, s )d s ≤ (cid:3) A.4. Concluding the proof of Theorem 3.4. It is not hard to transform the L statementof Proposition A.2 into the desired “almost sure” statement: Proposition A.8. For any ε > , t ≥ , w.h.p max x ∈{− L,...,L +1 } L | Φ( x, L t ) − ˆ h x ( L t ) | ≤ ε. (A.50) Proof. Also here we write h for ˆ h . Note that from Lemma A.3 ( iii ) the random vector ( | η x ( t ) | ) x ∈{− L,...,L } is stochastically dominated for every t by a vector of IID time-independent geometric variables.This implies that there exists a constant C such that for any t ≥ 0, w.h.p. | h x ( t ) − h y ( t ) | ≤ C | x − y | for every x, y ∈ {− L, . . . , L } , | x − y | ≥ log L (A.51)(this can be proved by using large deviation estimates and a union bound on x, y ∈ {− L, . . . , L } ).Moreover Lemma A.3 ( i ) ensures that Φ( · , t ) is always Lipschitz so that (A.51) holds also forΦ( · , t ) − h · ( t ).With (A.51) and L large enough, one has (cid:26) max x ∈{− L,...,L +1 } | Φ( x, L t ) − h x ( L t ) | ≥ εL (cid:27) ⊂ X x ∈{− L,...,L +1 } | Φ( x, L t ) − h x ( L t ) | ≥ ε L C (A.52)so that the left-hand side event has small probability when L is large, otherwise Proposition A.2would be false. (cid:3) A.5. Laplacian bounds. Recall that Φ( x, t ) is the solution of the Cauchy problem (3.12). Wewant to bound Φ( x, t ) above and below with the solution of a suitable heat equation. For this,we will suppose that the function φ , through which the initial condition Φ for Φ( x, t ) is defined,is concave on [ − , 1] (in addition to the assumptions required for Theorem 3.4). One defines theevolution Φ ( x, t ) as the solution of ∂ t Φ ( x, t ) = ∆Φ ( x, t )Φ ( − L, t ) = Φ( L + 1 , t ) = 0Φ ( x, 0) = Φ ( x ) (A.53)for t ≥ , x ∈ {− L +1 , L } . Also we define Φ ( x, t ) as the solution of the analogous equation (withthe same boundary values) where the discrete Laplacian is multiplied by (1 / σ ′ ( k ∂ x φ k ∞ ) =1 / (1 + k ∂ x φ k ∞ ) . Proposition A.9. For every t ≥ every x ∈ {− L, . . . , L + 1 } one has Φ ( x, t ) ≤ Φ( x, t ) ≤ Φ ( x, t ) . (A.54) Proof. We prove the upper bound, the lower one being very similar. Suppose that the resultdoes not hold and set T := max { t | Φ( x, t ) ≤ Φ ( x, t ) for every t ≤ T, x ∈ {− L, . . . , L + 1 }} . (A.55)Note that by property of the heat-equation, Φ ( x, t ) is a strictly concave function of x for allpositive t (except in the case where one starts from the flat initial condition but in that case thestatement is trivial). Let x be such thatΦ( x , T ) = Φ ( x , T ) . (A.56)Then one remarks that q x ( T ) − q x − ( T ) < ( · , T )) and that byLemma A.3 max x | q x ( t ) | ≤ k ∂ x φ k ∞ so that σ ( q x ( T )) − σ ( q x − ( T )) < ( q x ( T ) − q x − ( T )) σ ′ ( k ∂ x φ k ∞ ) (A.57)(since σ ′ ( · ) is decreasing on R + ) and hence2 ∂ t [Φ − Φ]( x , T ) = σ ′ ( k ∂ x φ k ∞ )∆Φ ( x, t ) − σ ( q x ( T )) + σ ( q x − ( T )) > σ ′ ( k ∂ x φ k ∞ )[(Φ ( x + 1 , t ) + Φ ( x − , t )) − (Φ( x + 1 , t ) + Φ( x − , t ))] . (A.58)Since the last expression is non-negative, one has Φ( x, t ) < Φ ( x, t ) on an interval [ T, T + ε ( x )]for some ε ( x ) > 0, for every x ∈ {− L, . . . , L + 1 } and that concludes the proof since the onlypossibility is that T = ∞ . 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