Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters
aa r X i v : . [ m a t h . P R ] F e b Slightly supercritical percolation on nonamenable graphs I:The distribution of finite clusters
Tom Hutchcroft
Statslab, DPMMS, University of Cambridge.Email: [email protected]
February 10, 2020
Abstract.
We study the distribution of finite clusters in slightly supercritical ( p ↓ p c ) Bernoullibond percolation on transitive nonamenable graphs, proving in particular that if G is a transitivenonamenable graph satisfying the L boundedness condition ( p c < p → ) and K denotes the clusterof the origin then there exists δ > P p ( n ≤ | K | < ∞ ) ≍ n − / exp (cid:20) − Θ (cid:16) | p − p c | n (cid:17)(cid:21) and P p ( r ≤ Rad( K ) < ∞ ) ≍ r − exp (cid:20) − Θ (cid:16) | p − p c | r (cid:17)(cid:21) for every p ∈ ( p c − δ, p c + δ ) and n, r ≥
1, where all implicit constants depend only on G . We deducein particular that the critical exponents γ ′ and ∆ ′ describing the rate of growth of the moments ofa finite cluster as p ↓ p c take their mean-field values of 1 and 2 respectively.These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, toproducts with trees, and to transitive graphs of spectral radius ρ < /
2. In particular, everyfinitely generated nonamenable group has a Cayley graph to which these results apply. They arenew for graphs that are not trees. The corresponding facts are yet to be understood on Z d evenfor d very large. In a second paper in this series, we will apply these results to study the geometricand spectral properties of infinite slightly supercritical clusters in the same setting. In Bernoulli bond percolation , each edge of a countable graph G = ( V, E ) is either deleted( closed ) or retained ( open ) independently at random with retention probability p ∈ [0 ,
1] toobtain a random subgraph ω of G . The connected components of ω are referred to as clusters . Wewill be primarily interested in the case that G is transitive , i.e., that the automorphism group of G acts transitively on V , or more generally that G is quasi-transitive , i.e., that the action of the1utomorphism group of G on V has at most finitely many orbits. When G is infinite, the criticalprobability p c = p c ( G ) is defined by p c = sup (cid:8) p ∈ [0 ,
1] : every cluster is finite P p -a.s. (cid:9) , where we write P p = P Gp for the law of Bernoulli- p bond percolation on G . It is now known thatthe phase transition is non-trivial (i.e., that 0 < p c <
1) for every infinite quasi-transitive graphwith superlinear volume growth [8, 16, 45].Percolation theorists are primarily interested in the geometry of clusters, and how this geometrychanges as p is varied. The theory is naturally decomposed into several regimes according to therelationship between p and p c . One possible taxonomy is as follows:1. The subcritical regime, in which 0 < p < p c .2. The slightly subcritical regime, in which 0 < p c − p ≪ critical regime, in which p = p c .4. The slightly supercritical regime, in which 0 < p − p c ≪ supercritical regime, in which p c < p < very subcritical regime p ≪ verysupercritical regime 1 − p ≪
1; these regimes are often much easier to understand.) Among allof these regimes, the most difficult to study is usually the slightly supercritical regime. A centraldifficulty in the study of this regime, and in supercritical percolation more generally, is that one isinterested in the probability of highly non-monotone events for the percolation configuration, suchas { n ≤ | K | < ∞} where K is the cluster of the origin, while many of the tools that have beendeveloped in the study of the other regimes are either mostly or exclusively suited to the analysisof monotone events and functions.Indeed, there are essentially only two examples in which slightly supercritical percolation isreasonably well understood: trees and site percolation on the triangular lattice. In both cases,there are exact duality relations, developed extensively in the Euclidean setting by Kesten [38], thatallow us to convert questions about slightly supercritical percolation into questions about slightly subcritical percolation. In the case of trees these slightly subcritical questions can then be answeredwith the classical theory of branching processes (see e.g. [25, Chapter 10]), while for site percolationon the triangular lattice Smirnov and Werner [57] showed that they can be answered by combining For k -regular trees and p ≥ p c , the conditional distribution of the cluster of the origin given that it is finite isthe same in Bernoulli- p and Bernoulli- q percolation, where q is the unique to solution to q (1 − q ) k − = p (1 − p ) k − lying in [0 , p c ]. This duality is a consequence of the fact that every finite connected subgraph of a k -regular treecontaining n edges also touches exactly ( k − n + k edges that it does not contain. As p ↓ p c , this dual probability q satisfies p c − q ∼ p − p c . Thus, all questions concerning the distribution of finite clusters in slightly supercriticalpercolation can immediately be converted into questions concerning slightly subcritical percolation, which are mucheasier. This property is very specific to trees, and these arguments do not generalize to other nonamenable transitivegraphs. Let us note, however, that slightly more involved duality arguments should also allow one to understandslightly supercritical percolation on transitive nonamenable proper plane graphs with locally finite planar dual; toour knowledge such an analysis has not been carried out in the literature. Note that such graphs are always Gromovhyperbolic [22] and therefore have p c < p → by the results of [36]. Thus, the results of this paper are alwaysapplicable to them. Z d remains poorly understood even when d isvery large and all other regimes are now understood rather thoroughly. Highlights of the literatureregarding the other regimes include [3, 20, 47] for the subcritical regime, [5, 27, 40, 41] for the criticaland slightly subcritical regimes, and [4, 15, 26, 39] for the supercritical regime. See e.g. [13, 25, 30]for overviews of this literature and of open problems in high dimensional percolation, and [17] forsome interesting recent partial progress on slightly supercritical percolation. Let us also mentionthat a good understanding of slightly supercritical percolation appears to be a prerequisite to thesolution of several important open problems regarding invasion percolation and minimal spanningforests, see [30, Section 16.1] and references therein.The primary purpose of this series of two papers is to study slightly supercritical percolationin the ‘infinite-dimensional’ setting of nonamenable (quasi-)transitive graphs. Here, we recall thata connected, locally finite graph is said to be nonamenable if its Cheeger constant Φ( G ) = inf (cid:26) | ∂ E W | P w ∈ W deg( w ) : W a finite set of vertices (cid:27) is positive, where ∂ E W denotes the set of edges with one endpoint in W and one endpoint notin W ; G is said to be amenable if it is not nonamenable, i.e., if its Cheeger constant is zero.Background on percolation in the nonamenable context may be found in e.g. [46]. We prove ourresults under the additional hypothesis that G satisfies the L boundedness condition , which wasintroduced in [36] and studied further in [34]. Let us now briefly introduce this condition. Given acountable graph G = ( V, E ), we write T p ( u, v ) = P p ( u ↔ v ) for the two-point matrix , and define p → = p → ( G ) = sup n p ∈ [0 ,
1] : k T p k → < ∞ o , where we recall that if M ∈ [0 , ∞ ] V is a V -indexed matrix with non-negative entries then the L ( V ) → L ( V ) operator norm k M k → ∈ [0 , ∞ ] is defined by k M k → = sup n k M f k : f ∈ L ( V ) , k f k = 1 o . We say that G satisfies the L boundedness condition if p c ( G ) < p → ( G ). This condition is conjec-tured to hold for every connected, locally finite, nonamenable quasi-transitive graph [34, Conjec-ture 1.3], and is now known to hold for several classes of examples, including Gromov hyperbolicgraphs [36], highly nonamenable graphs [48,52,54], and graphs admitting a quasi-transitive nonuni-modular subgroup of automorphisms [33]. In particular, it can be deduced by the methods of [52]that every nonamenable, finitely generated group has a Cayley graph for which p c < p → . (Onthe other hand, we always have that p c = p → in the amenable case.) See [34] for an overview.See also [7, 37] and references therein for an overview of what is known regarding critical andnear-critical percolation on general nonamenable transitive graphs without this assumption.The main results of this paper apply the L boundedness condition to establish a very precise3nderstanding of the distribution of finite clusters in critical and near critical percolation. Ina forthcoming sequel to this paper [32], we will then apply these results to study the large-scalegeometry of infinite clusters in slightly supercritical percolation. All of our results regarding slightlysupercritical percolation are new when the graph in question is not a tree.The results of both papers build upon the methods of our recent work with Hermon [29], whichestablished related, non-quantitative results for supercritical percolation on nonamenable transitivegraphs (that do not necessarily satisfy the L boundedness condition). Making these argumentsquantitative in a sharp way in order to get the correct behaviour as p ↓ p c is a surprisingly delicatematter, and our proofs are, unfortunately, substantially more technical than those of [29].Besides the intrinsic interest of our results, we are also hopeful that some of the tools wedevelop will be useful for approaching the high-dimensional Euclidean case; some perspectives onthe remaining challenges in this case are presented in Section 5. It would also be very interesting(and seemingly highly non-trivial) to extend our methods to other infinite-dimensional settings, suchas hypercubes or expander graphs (which are finite analogues of nonamenable graphs). Critical andslightly subcritical percolation on these graphs has been studied in many works, surveyed in [58],the highlights of which include [10–12, 31, 59]. (The analogous results for the complete graph areclassical, see [9] and references therein.) We now state our results concerning the distribution of finite clusters in near critical percolation.While the supercritical aspects of these results are the most novel, it seems that they also improveslightly upon the best existing estimates for slightly subcritical percolation. We write K v for thecluster of v and | K v | for the number of vertices it contains. Theorem 1.1 (Volume of finite clusters) . Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c ( G ) < p → ( G ) . Then there exists a constant δ = δ ( G ) > suchthat P p (cid:0) n ≤ | K v | < ∞ (cid:1) ≍ n − / exp (cid:20) − Θ (cid:16) | p − p c | n (cid:17)(cid:21) (1.1) for every n ≥ , v ∈ V , and p ∈ ( p c − δ, p c + δ ) , where all implicit constants depend only on G . Here and below, we write ≍ , (cid:23) , and (cid:22) to denote equalities and inequalities that hold up topositive multiplicative constants depending only on the graph G . Thus, for example, “ f ( n ) ≍ g ( n )for every n ≥
1” means that there exist positive constants c and C such that cg ( n ) ≤ f ( n ) ≤ Cg ( n ) for every n ≥
1. We use Landau’s asymptotic notation similarly, so that, for example, f ( n ) = Θ( g ( n )) if and only if f ≍ g , and f ( n ) (cid:22) g ( n ) if and only if f ( n ) = O ( g ( n )). In particular,Theorem 1.1 is equivalent to the assertion that there exist positive constants c , c , C , C , and δ such that c n − / exp h − C | p − p c | n i ≤ P p (cid:0) n ≤ | K v | < ∞ (cid:1) ≤ C n − / exp h − c | p − p c | n i for every v ∈ V , p ∈ ( p c − δ, p c + δ ), and n ≥
1. 4ur next theorem establishes a similar result for the radius of a finite supercritical cluster. Wewrite Rad int ( K v ) and Rad ext ( K v ) for the intrinsic and extrinsic radii of K v , that is, the maximumdistance from v to another point of K v in the graph metric on K v and in the graph metric on G respectively. Note that we trivially have Rad ext ( K v ) ≤ Rad int ( K v ). Theorem 1.2 (Radii of finite clusters) . Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c ( G ) < p → ( G ) . Then there exists a constant δ = δ ( G ) > suchthat P p (cid:0) r ≤ Rad int ( K v ) < ∞ (cid:1) ≍ r − exp (cid:20) − Θ (cid:16) | p − p c | r (cid:17)(cid:21) (1.2) and P p (cid:0) r ≤ Rad ext ( K v ) < ∞ (cid:1) ≍ r − exp (cid:20) − Θ (cid:16) | p − p c | r (cid:17)(cid:21) (1.3) for every r ≥ , v ∈ V , and p ∈ ( p c − δ, p c + δ ) , where all implicit constants depend only on G . The parts of these results concerning critical percolation were already known, and are appliedas a component of the proof. Indeed, a connected, locally finite, quasi-transitive graph G is said tosatisfying the triangle condition if ∇ p c ( v ) := X u,w ∈ V T p ( v, u ) T p ( u, w ) T p ( w, v ) < ∞ for every v ∈ V . The triangle condition was introduced by Aizenman and Newman [5] and provento hold on Z d with d large in the groundbreaking work of Hara and Slade [27]. It is conjecturedto hold if and only if d >
6, and is now known to hold for all d ≥
11 [23]. It is known that if aconnected, locally finite, quasi-transitive graph G satisfies the triangle condition then P p c ( | K v | ≥ n ) ≍ n − / for every n ≥ v ∈ V , and that (1.4) P p c (Rad int ( K v ) ≥ r ) (cid:22) r − for every r ≥ v ∈ V , (1.5)so that, in particular, every cluster is finite P p c -almost surely. Note that the triangle condition isequivalent to the assertion that T p c ( v, v ) < ∞ for every v ∈ V , and is therefore implied by the L boundedness condition since T p ( v, v ) ≤ k T p k → ≤ k T p k → . The upper and lower bounds of (1.4)follow from the work of Aizenman and Newman [5] and Aizenman and Barsky [3] respectively, while(1.5) follows from the work of Kozma and Nachmias [40]. A simple proof of the complementarylower bound P p c (Rad int ( K v ) ≥ r ) (cid:23) r − , which holds on every connected, locally finite, quasi-transitive graph, is given in Proposition 4.2. Moreover, in [34] it is shown that the L boundednesscondition allows one to compare intrinsic and extrinsic distances, which allows one to prove inparticular that P p c (Rad ext ( K v ) ≥ r ) ≍ r − for every r ≥ v ∈ V . (1.6)for every connected, locally finite, quasi-transitive graph G satisfying the L boundedness condi-tion. (On the other hand, Kozma and Nachmias [41] proved that P p c (Rad ext ( K v ) ≥ r ) ≍ r − for5ercolation on Z d with d large. The disparity between these two results is related to the fact thatrandom walk is diffusive on Z d and ballistic on nonamenable graphs.)Let E ( K v ) be the set of edges that touch (i.e., have at least one endpoint in) K v , and define ζ ( p ) = − lim sup n →∞ n log P p (cid:0) n ≤ | E ( K v ) | < ∞ (cid:1) (1.7)to be the exponential rate of decay of the probability that v belongs to a large finite cluster(which is easily seen not to depend on the choice of v ). It is a consequence of the sharpness ofthe phase transition that ζ ( p ) > every connected, locally finite, quasi-transitive graph G and every 0 ≤ p < p c . This was first proven by Aizenman and Barsky [3] and Aizenman andNewman [5] (see also the closely related work of Menshikov [47]), and several alternative proofsare now available [19, 20, 35]. On the other hand, for supercritical percolation on quasi-transitivegraphs, it is shown in [29] that p c < ζ ( p ) > some p c < p < p c < ζ ( p ) > every p c < p <
1, if and only if G is nonamenable. Thus, for connected, locally finite,nonamenable quasi-transitive graphs, we have that ζ ( p ) > p = p c . Note howeverthat these arguments do not give any quantitative control on the manner in which ζ ( p ) → p → p c . Theorem 1.1 provides such a quantitative understanding, and yields in particular thefollowing immediate corollary. Corollary 1.3.
Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c ( G ) < p → ( G ) . Then there exists δ > such that ζ ( p ) ≍ | p − p c | for every p ∈ ( p c − δ, p c + δ ) . Of course, Theorems 1.1 and 1.2 tell us rather more than this: they show us the precise mannerin which the polynomial tail at p c is gradually transformed into the exponential tail away from p c .In particular, they make the following natural heuristic picture precise: There is a scaling window of order | p − p c | − such that within the scaling window percolation behaves in essentially the sameway as critical percolation, whereas outside the scaling window the off-critical effects begin tobecome apparent. Moreover, roughly speaking, these off-critical effects manifest themselves in away that is proportional to how much larger our cluster is than a cluster that is at the edge of thescaling window (i.e., than a cluster that has radius | p − p c | − or volume | p − p c | − ). This intuitivepicture will be an important motivation to many of our proofs: We will often prove estimates byseparate analyses of the ‘inside-window’ and ‘outside-window’ cases. Note that the restriction to aneighbourhood of p c is necessary as ζ ( p ) → ∞ as p ↓ p ↑ γ ′ and ∆ ′ . It is believed that for every connected, locally finite,quasi-transitive graph G = ( V, E ) there exists γ, γ ′ , ∆ , and ∆ ′ such that E p h | K v | k i ≍ k | p − p c | − γ − ( k − ± o k (1) as p ↑ p c and (1.8) E p h | K v | k ( | K v | < ∞ ) i ≍ k | p − p c | − γ ′ − ( k − ′ ± o k (1) as p ↓ p c , (1.9)where the k subscripts mean that the implicit constants may depend on k . See [25, Chapters 9and 10] for background on this conjecture. It is known that if G satisfies the triangle conditionthen γ and ∆ are well-defined and take their mean-field values of 1 and 2 respectively [5, 50]6see also [35]). Theorem 1.1 implies a similar result for γ ′ and ∆ ′ for graphs satisfying the L boundedness condition. Corollary 1.4.
Let G be a connected, locally finite, quasi-transitive graph such that p c ( G )
Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph, and let p c = p c ( G ) . Suppose that ∇ p c < ∞ . Then there exist positive constants δ and C such that the bounds P Hp ( R v ≥ r ) ≤ C (cid:18) r ∨ ( p − p c ) (cid:19) and P Hp ( E v ≥ n ) ≤ C (cid:18) n / ∨ ( p − p c ) (cid:19) hold for every r, n ≥ , every p ∈ [0 , p c + δ ) , every subgraph H of G , and every vertex v of H . We stress that in the statement and proof of this estimate, p c always refers to p c ( G ). The proofwill make use of Russo’s formula [25, Theorem 2.32], which states that if X : { , } E → R dependson at most finitely many edges then E p (cid:2) X ( ω ) (cid:3) is a polynomial in p with derivative ddp E p (cid:2) X ( ω ) (cid:3) = X e ∈ E E p (cid:2) X ( ω e ) − X ( ω e ) (cid:3) = 1 p X e ∈ E E p h ( ω ( e ) = 1) (cid:0) X ( ω ) − X ( ω e ) (cid:1)i for every p ∈ [0 , ω e = ω ∪ { e } and ω e = ω \ { e } . We write a ∨ b := max { a, b } and a ∧ b := min { a, b } . We also write B int ( v, n ) for the intrinsic ball of radius n around v in K v , andwrite ∂B int ( v, n ) = B int ( v, n ) \ B int ( v, n −
1) for the set of vertices at intrinsic distance exactly n from v . 9 roof of Lemma 2.1. Fix H and v . We know by the results of [40, 53] (see also [33, Section 6]) that P Hp ( R v ≥ r ) (cid:22) r − and E Hp (cid:2) B int ( v, r ) (cid:3) (cid:22) r for every 0 ≤ p ≤ p c . Observe that, for each r ≥
1, if K v has intrinsic radius at least r and e is suchthat K v ( ω e ) does not have intrinsic radius at least r , then e must lie on every intrinsic geodesic oflength r starting at v in K v . There are clearly at most r such edges, and it follows from Russo’sformula that ddp P Hp ( R v ≥ r ) ≤ rp P Hp ( R v ≥ r ) ≤ rp c P Hp ( R v ≥ r )for every p c ≤ p ≤ r ≥
1. This inequality may be written equivalently as ddp log P Hp ( R v ≥ r ) ≤ rp c . Integrating this bound between p and p c yields that P Hp ( R v ≥ r ) ≤ P Hp c ( R v ≥ r ) exp (cid:20) ( p − p c ) rp c (cid:21) (cid:22) r exp (cid:20) ( p − p c ) rp c (cid:21) (2.1)for every p c ≤ p ≤ r ≥
1. Since P Hp ( R v ≥ r ) is decreasing in r , it follows that P Hp ( R v ≥ r ) (cid:22) min ( ℓ exp (cid:20) ( p − p c ) p c ℓ (cid:21) : 1 ≤ ℓ ≤ r ) for every p c ≤ p ≤ r ≥
1. The claimed bound on the tail of the intrinsic radius follows bytaking ℓ = r ∧ ⌈ ( p − p c ) − ⌉ .Now, a similar argument to above yields that ddp log E Hp (cid:2) B int ( v, r ) (cid:3) ≤ rp c for every p c ≤ p ≤ r ≥
1, and hence that E Hp (cid:2) B int ( v, r ) (cid:3) (cid:22) r exp (cid:20) p − p c p c r (cid:21) (2.2)for every p c ≤ p ≤ r ≥
1. It follows by the union bound and Markov’s inequality that P Hp ( E v ≥ n ) ≤ n E Hp (cid:2) B int ( v, r ) (cid:3) + P Hp ( R v ≥ r ) (cid:22) rn exp (cid:20) p − p c p c r (cid:21) + (cid:20) r ∨ ( p − p c ) (cid:21) for every n, r ≥
1. The claim follows by taking r = ⌈ n / ∧ ( p − p c ) − ⌉ . In this section we prove the upper bounds of Theorems 1.1 and 1.2 in the case p > p c .10 .1 Setting up the main differential inequalities Most of the the work to prove Theorems 1.1 and 1.2 will concern the case that p > p c is slightlysupercritical and n and r are outside the scaling window, so that either n ≫ | p − p c | − or r ≫| p − p c | − . As discussed above, we follow the basic strategy of [29], but apply the assumption that p c < p → to make the proof quantitative. We begin by recalling some notation from [29]. Let G = ( V, E ) be a connected, locally finite, transitive, nonamenable graph, and let v be a vertex of G . Let K v denote the cluster of v , and let E v = | E ( K v ) | be the number of edges touching K v .Define H to be the set of all finite connected subgraphs of G , and let H v be the set of all finiteconnected subgraphs of G containing v . Given a function F : H v → R , we write E p,n [ F ( K v )] := E p (cid:2) F ( K v ) ( E v ≤ n ) (cid:3) and E p, ∞ [ F ( K v )] := E p (cid:2) F ( K v ) ( E v < ∞ ) (cid:3) for every p ∈ [0 ,
1] and n ≥ F : H v → R and n ≥
1, Russo’s formula allows us to express the derivative of thetruncated expectation E p,n [ F ( K v )], which is a polynomial in p , in terms of pivotal edges andobtain that ddp E p,n (cid:2) F ( K v ) (cid:3) = U p,n [ F ( K v )] − D p,n [ F ( K v )] (3.1)where we write U p,n (cid:2) F ( K v ) (cid:3) := 1 p X e ∈ E E p,n (cid:20)(cid:16) F [ K v ] − F (cid:2) K v ( ω e ) (cid:3)(cid:17) (cid:0) ω ( e ) = 1 (cid:1)(cid:21) and D p,n (cid:2) F ( K v ) (cid:3) := 11 − p X e ∈ E E p h F ( K v ) (cid:0) ω ( e ) = 0 , E v ≤ n < E v ( ω e ) (cid:1)i . See [29, Section 2] for further details. Intuitively, in the n → ∞ limit, the term D p, ∞ (cid:2) F ( K v ) (cid:3) accounts for the effect of finite clusters becoming infinite, while the term U p, ∞ (cid:2) F ( K v ) (cid:3) accountsfor the effect of finite clusters growing while remaining finite. (Note however that the above formulasare only a priori valid for finite n .) Note that U p,n [ F ( K v )] is non-negative if F is increasing andthat D p,n [ F ( K v )] is non-negative if F is non-negative. Note also that U p,n [ F ( K v )] and D p,n [ F ( K v )]both depend linearly on the function F .In order to prove Theorems 1.1 and 1.2, we will need to prove lower bounds on D p,n [ F ( K v )]and upper bounds on U p,n [ F ( K v )] for appropriate choices of F . The two quantities will often haveroughly the same order, making the analysis of their difference rather delicate. In this section we prove a lower bound on D p,n [ F ( K v )] for non-negative F . In [29, Proposition2.1], it is shown via an ineffective argument that if G is transitive and nonamenable then for every p c < p ≤ c p such that D p,n (cid:2) F ( K v ) (cid:3) ≥ c p E p,n (cid:2) | K v | · F ( K v ) (cid:3) p ≤ p ≤ F : H v → [0 , ∞ ). A key ingredient to the proofof our main theorems is the following proposition, which allows us to take c p of order ( p − p c ) underthe assumption that p c < p → . We write θ ∗ ( p ) = inf v ∈ V P p ( v → ∞ ) and θ ∗ ( p ) = sup v ∈ V P p ( v →∞ ). Proposition 3.1.
Let G be a countable graph. Then D p,n (cid:2) F ( K v ) (cid:3) ≥ " θ ∗ ( p ) p (1 − p ) M θ ∗ ( p ) k T p k → θ ∗ ( p ) E p,n (cid:2) | K v | · F ( K v ) (cid:3) (3.2) for every non-negative function F : H v → [0 , ∞ ) , every n ≥ , and every p ∈ [0 , . Consequently,if G is connected, locally finite, and quasi-transitive with p c ( G ) < p → ( G ) , then there exist positiveconstants δ > and c > such that D p,n (cid:2) F ( K v ) (cid:3) ≥ c ( p − p c ) E p,n (cid:2) | K v | · F ( K v ) (cid:3) (3.3) for every non-negative function F : H v → [0 , ∞ ) , every n ≥ , and every p ∈ ( p c , p c + δ ) . The precise form of the argument given below was suggested to us by Antoine Godin; a similarargument will appear in his forthcoming PhD thesis [24]. We thank him for sharing this argumentwith us, which substantially simplified our proof.The proof makes use of the notion of the BK inequality and the associated notion of the disjointoccurence A ◦ B of two events A and B ; We refer the unfamiliar reader to [25, Chapter 2.3] forbackground. Proof of Proposition 3.1.
Let G be a countable graph. For each vertex v of G , let E → v denote theset of oriented edges e of G with e − = v . We first claim that for each deterministic finite set ofvertices S ⊆ V we have thatΨ p ( S ) := 11 − p X v ∈ S X e ∈ E → v ( e + / ∈ S ) P p ( e + → ∞ off S ) ≥ " θ ∗ ( p ) p (1 − p ) M θ ∗ ( p ) k T p k → θ ∗ ( p ) | S | (3.4)for every 0 < p < p → . (Note that we have written the expression on the right in this way asthe bracketed term is of constant order in cases of interest.) The deduction of (3.2) from (3.4) isidentical to the proof of [29, Proposition 2.1] and is omitted. Indeed, the proof of [29, Proposition2.1] shows more generally that D p,n (cid:2) F ( K v ) (cid:3) ≥ E p,n (cid:2) F ( K v )Ψ p ( K v ) (cid:3) for every p ∈ [0 , n ≥
1, and every non-negative F : H v → [0 , ∞ ).Let S be a deterministic finite set of vertices. Let ∂ → E S denote the set of oriented edges of G with e − ∈ S and e + / ∈ S . Observe that for each u ∈ S we have that {| K u | = ∞} ⊆ [ e ∈ ∂ → E S { u ↔ e − } ◦ { e open } ◦ { e + → ∞ off S } . Indeed, suppose that u ∈ S is in an infinite cluster, and let γ be an infinite simple open path12tarting at u . Since S is finite, there is some last vertex v of S that is visited by γ . Let e be theedge of ∂ → E S that is crossed by γ as it leaves v , which is necessarily open. Then the pieces of γ before and after crossing e are disjoint witnesses for the events { u ↔ e − } and { e + → ∞ off S } ,both of which are disjoint from the edge e . Thus, applying the BK inequality and the union boundyields that P p ( u → ∞ ) ≤ p X v ∈ S T p ( u, v ) X e ∈ E → v ( e + / ∈ S ) P p ( e + → ∞ off S )for every u ∈ S . Summing over u we obtain that | S | θ ∗ ( p ) ≤ p X v ∈ S X u ∈ S T p ( u, v ) X e ∈ E → v ( e + / ∈ S ) P p ( e + → ∞ off S ) . (3.5)Define f : V → R by f p ( v ) = 11 − p ( v ∈ S ) X e ∈ E → v ( e + / ∈ S ) P p ( e + → ∞ off S ) . Rewriting the above inequality (3.5) in terms of f and applying Cauchy-Schwarz, we obtain that θ ∗ ( p ) | S | p (1 − p ) ≤ h T p S , f i ≤ k T p k → k S k k f k ≤ k T p k → | S | / k f k / k f k / ∞ , and since we clearly have that k f k ∞ ≤ M θ ∗ ( p ) / (1 − p ), it follows that11 − p X v ∈ S X e ∈ E → v ( e + / ∈ S ) P p ( e + → ∞ off S ) = k f k ≥ " θ ∗ ( p ) p (1 − p ) M θ ∗ ( p ) k T p k → θ ∗ ( p ) | S | as claimed.The deduction of (3.3) from (3.2) follows by standard arguments: Indeed, if G is connectedand quasi-transitive then there exists C such that θ ∗ ( p ) ≥ p C θ ∗ ( p ) for every p ∈ [0 , p c ( G ) < p → ( G ) then k T p k → is bounded on a neighbourhood of p c . On the other hand, forquasi-transitive graphs there always exists a positive constant c such that θ ∗ ( p ) ≥ c ( p − p c ) for all p c ≤ p ≤ Remark . The proof of [29, Proposition 2.1] can also be made quantitative under the assumptionthat p c < p → , since in this case we know that the density of trifurcations is of order ( p − p c ) [34,Corollary 5.6]. Note however that the resulting bound is not sharp.An easy corollary of Proposition 3.1 is the following weak version of the first moment estimatefrom Corollary 1.4. This weak estimate will nevertheless be useful to us as boundary data whenwe analyze a certain differential inequality later in the paper. Corollary 3.3.
Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c < p → . Then there exist positive constants δ and C such that inf n ( p − p c ) E p, ∞ | K v | : p ∈ ( p c + ε, p c + 2 ε ) o ≤ C for every v ∈ V and < ε ≤ δ roof. Fix v ∈ V . Since G is quasi-transitive and satisfies the triangle condition, there exists aconstant C such that P p ( | K v | = ∞ ) ≤ C ( p − p c ) for every p c ≤ p ≤
1. On the other hand,Proposition 3.1 implies that there exist positive constants c and δ such that ddp P p ( | K v | > n ) = − ddp E p,n [1] = D p,n [1] ≥ c ( p − p c ) E p,n | K v | for every n ≥ p ∈ ( p c , p c + δ ]. Integrating this differential inequality yields that Z p c +2 εp c + ε c ( p − p c ) E p,n | K v | d p ≤ P p c +2 ε ( | K v | > n ) − P p c + ε ( | K v | > n ) ≤ P p c +2 ε ( | K v | > n )for every 0 < ε ≤ δ/
2. Using the monotone convergence theorem to take the limit as n → ∞ , weobtain that Z p c +2 εp c + ε c ( p − p c ) E p, ∞ | K v | d p ≤ Cε for every 0 < ε ≤ δ/
2. This is easily seen to imply the claim.
The goal of this section is to prove the following proposition, which establishes the upper boundsof Theorem 1.2. This is substantially easier than the corresponding upper bounds on the tail ofthe volume.
Proposition 3.4.
Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c < p → . Then there exist positive constants δ , c , and C such that P p ( r ≤ Rad ext ( K v ) < ∞ ) ≤ P p ( r ≤ Rad int ( K v ) < ∞ ) ≤ Cr − e − c ( p − p c ) r for every r ≥ , v ∈ V , and p ∈ [ p c , p c + δ ) . We begin with the following proposition, which upper bounds the probability of having a large skinny cluster, whose radius is large but whose volume is smaller than it should be given the largeradius. In particular, this proposition applies the assumption ∇ p c < ∞ to give a quantitativeimprovement to [29, Lemma 2.8]. This proposition will be extremely useful to us, and will beapplied many times throughout the paper. Proposition 3.5 (Skinny clusters) . Let G = ( V, E ) be a connected, locally finite, quasi-transitivegraph such that ∇ p c < ∞ . There exist positive constants δ , c , and C such that the bound P Hp ( r ≤ R v < ∞ and E v ≤ αR v ) ≤ C inf ((cid:18) r + λ (cid:19) exp h − ce − Cλα λr i : 0 ∨ ( p − p c ) ≤ λ ≤ δ ) holds for every ≤ p ≤ p c + δ , α ≥ , r ≥ , subgraph H of G , and vertex v of H . Here, we recall that E v denotes the number of edges of H touched by the percolation cluster of v in H , and R v denotes the intrinsic radius of this cluster. Again, we stress that in the statementand proof of this proposition, p c will always denote p c ( G ) and all implicit constants will dependonly on G . 14 roof of Proposition 3.5. Fix a subgraph H of G , a vertex v of H , 0 ≤ p ≤ r ≥ α ≥ λ ≥ ( p − p c ) ∨
0, let n = ⌈ /λ ⌉ + 2, and let k = ⌊ r/ n ⌋ −
1. It suffices to prove that there existpositive constants δ , c and C depending only on G such that if λ ≤ δ then P Hp ( r ≤ R v < ∞ and E v ≤ αR v ) (cid:22) (cid:18) r + λ (cid:19) exp h − ce − Cλα λr i . (3.6)The case r = O ( n ) of this inequality may be deduced easily from Lemma 2.1: Indeed, it followsfrom Lemma 2.1 that there exists δ > ≤ p ≤ p c + δ and r ≤ n then P Hp ( r ≤ R v < ∞ and E v ≤ αR v ) ≤ P Hp ( R v ≥ r ) (cid:22) r . (3.7)Moreover, we have that P H ′ p ( R w ≥ n − (cid:22) n exp (cid:20) λ ( n − p c (cid:21) (cid:22) λ (3.8)for every 0 ≤ p ≤ p c + δ , every n ≥
1, every subgraph H ′ of G and every vertex w of H ′ , where westress that, as always, the implicit constants depend only on G . The bound (3.7) is already of thedesired order when r ≤ n , since the quantity in the exponential on the right hand side of (3.6) isbounded in this regime. Thus, it suffices to prove that there exist positive constants δ , c , and C depending only on G such that if λ ≤ δ then P Hp ( r ≤ R v < ∞ and E v ≤ αR v ) (cid:22) λ exp h − ce − Cλα λr i (3.9)for every r ≥ n .To this end, suppose that r ≥ n , so that k ≥
1. Suppose further that 0 ≤ p ≤ p c + δ andthat 0 ≤ λ ≤ δ . Consider exploring the cluster of v as follows: at stage i , expose the value ofthose edges that touch ∂B int ( v, i − i − v , and have not yet been exposed. Stop when ∂B int ( v, i ) = ∅ . For each ℓ ≥
0, let X i be the setof edges whose status is queried at stage i , so that | X i | > ≤ i ≤ r on the event that R v ≥ r . Define a sequence of stopping times ( T j ) j ≥ for this exploration process by setting T = 0and recursively setting T j +1 = inf n i ≥ T j + n : 0 < | X i | ≤ α o , letting T j +1 = ∞ if the set on the right hand side is empty. We claim that T k < ∞ on the event that r ≤ R v < ∞ and E v ≤ αR v . Indeed, suppose that this event holds. Let k ′ = k ′ ( K v ) = ⌊ R v / n ⌋ − k ′ ≥ k ≥ k ′ ≥ R v / n . We trivially have that 2 nk ′ + n − n ( ⌊ R v / n ⌋− n − ≤ R v and that k ′ X a =1 n − X b =0 | X an + b | ≤ R v X i = n | X i | ≤ E v ≤ αR v , and it follows that there exists 0 ≤ b = b ( K v ) ≤ n − P k ′ a =1 | X an + b | ≤ αR v /n . ApplyingMarkov’s inequality, we deduce that there exists a subset A = A ( K v ) of { , . . . , k ′ } such that | A | ≥ k ′ and | X an + b ( K v ) | ≤ αR v /nk ′ ≤ α for every a ∈ A . If we enumerate A in increasing orderas A = { a , a , . . . } , then an easy induction shows that T i ≤ a i n + b < ∞ for every i ≤ k ′ and hence15or every i ≤ k as claimed.Let F i be the σ -algebra generated by the first i steps of the exploration process, and let F T i bethe stopped σ -algebra associated to the stopping time T i . We clearly have that P Hp ( T < ∞ | F T ) = P Hp ( T < ∞ ) ≤ P Hp ( R v ≥ n ) (cid:22) λ, where the final inequality follows from (3.7). Now let i ≥ F T i . If T i = ∞ thenwe trivially have that T i +1 = ∞ also. Now suppose that T i < ∞ . Enumerate the edges of X T i by X T i = { e , . . . , e ℓ } . For each 1 ≤ j ≤ ℓ , let e j be oriented so that e − j has intrinsic distance atmost T i − v . Let H be the subgraph of H spanned those edges that have not been queriedby time T i + 1 (i.e., those edges not in S T i j =0 X j ). Let K be the cluster of e +1 in H , and let H be the subgraph of H defined by deleting every edge that touches K from H . Inductively, foreach 2 ≤ j ≤ ℓ let K j be the cluster of e + j in H j − and let H j be the subgraph of H j − formed bydeleting every edge that touches K j from H j − . In order for T i +1 to be finite, we must have thatthere exists 1 ≤ j ≤ ℓ such that e j is open and that K j has radius at least n −
1. It follows from(3.8) that there exists a constant C such that for each 1 ≤ j ≤ ℓ , the conditional probability that e j is open, that e j does not touch K i for any 1 ≤ i < j , and that K j has radius at least n − F T i and the clusters K , . . . , K j − is at most Cλ , and hence that P Hp ( T i +1 < ∞ | F T i ) ≤ ( T i < ∞ ) h − | − ∧ Cλ | X Ti i ≤ ( T i < ∞ ) h − | − ∧ Cλ | α i . Taking products and using the bound 1 − x ≤ e − x , we obtain that P Hp ( T i < ∞ ) (cid:22) λ h − | − ∧ Cλ | α i i − ≤ λ exp h −| − ∧ Cλ | α ( i − i for every i ≥ P Hp ( R v ≥ r, E v ≤ αR v ) ≤ P Hp ( T k < ∞ ) (cid:22) λ exp h −| − ∧ Cλ | α ( k − i . Applying the bound 1 − x ≤ e − x a second time, it follows that there exist positive constants δ , c and C ′ such that if λ ≤ δ and r ≥ n then P Hp ( R v ≥ r, E v ≤ αR v ) (cid:22) λ exp h − ce − C ′ λα λr i . (3.10)The proof may be concluded by combining the bounds (3.7) and (3.10), which hold for r ≤ n and r ≥ n respectively. Remark . The expression e − Cλα λ is maximized by λ = 1 /Cα . In particular, taking α = rs and λ = 1 /rs , it follows from Proposition 3.5 and Proposition 4.2 that, under the hypotheses of thoseresults, there exist constants c and C such that P p c (cid:0) E v ≤ s − r | R v ≥ r (cid:1) ≤ Ce − cs (3.11)for every v ∈ V , r ≥
1, and s ≥ roof of Proposition 3.4. The first inequality is trivial, so it suffices to prove the second. It followsfrom Proposition 3.1 that there exist positive constants δ and c such that ddp P p,n ( R v ≥ r ) ≤ − c ( p − p c ) E p,n (cid:2) E v ( R v ≥ r ) (cid:3) + U p,n (cid:2) ( R v ≥ r ) (cid:3) for every r ≥ n ≥ p ∈ [ p c , p c + δ ). As in the proof of Lemma 2.1, we can bound p U p,n (cid:2) ( R v ≥ r ) (cid:3) by the expected number of open edges e such that the cluster of v has intrinsicradius at least r in ω and strictly less than r in ω e . Since any such open edge must lie on everyintrinsic geodesic of length r starting from v in ω , we deduce that ddp P p,n ( R v ≥ r ) ≤ − c ( p − p c ) E p,n (cid:2) E v ( R v ≥ r ) (cid:3) + rp c P p,n ( R v ≥ r ) (3.12)for every r ≥ n ≥ p ∈ [ p c , p c + δ ).On the other hand, it follows from Proposition 3.5 that there exists positive constants δ , c , c , C , and C such that P p, ∞ (cid:16) R v ≥ r ≥ c p c ( p − p c ) E v (cid:17) ≤ C (cid:20) r + ( p − p c ) (cid:21) exp h − c e − C ( p − p c )[ c p c ( p − p c )] − ( p − p c ) r i ≤ C (cid:20) r + ( p − p c ) (cid:21) exp (cid:2) − c ( p − p c ) r (cid:3) ≤ C r exp (cid:2) − c ( p − p c ) r (cid:3) for every r ≥ p ∈ [ p c , p c + δ ), where we used that xe − x ≤ e − x − for every x ≥ c ( p − p c ) E p,n (cid:2) E v ( R v ≥ r ) (cid:3) ≥ rp c P p,n ( R v ≥ r, c ( p − p c ) E v ≥ r ) ≥ rp c P p,n ( R v ≥ r ) − rp c P p, ∞ (cid:16) R v ≥ r ≥ c p c ( p − p c ) E v (cid:17) ≥ rp c P p,n ( R v ≥ r ) − C p c exp (cid:2) − c ( p − p c ) r (cid:3) for every r ≥ p ∈ [ p c , p c + δ ). Letting δ = δ ∧ δ , we deduce from this and (3.12) that ddp P p,n ( R v ≥ r ) ≤ − rp c P p,n ( R v ≥ r ) + 2 C p c exp (cid:2) − c ( p − p c ) r (cid:3) for every r ≥ p ∈ [ p c , p c + δ ). Letting c = c ∧ (1 /p c ) and C = 2 C /p c , it follows that ddp h e c ( p − p c ) r P p,n ( R v ≥ r ) i ≤ (cid:20) c r − rp c (cid:21) e c ( p − p c ) r P p,n ( R v ≥ r ) + 2 C p c e ( c − c )( p − p c ) r ≤ C p c e ( c − c )( p − p c ) r ≤ C . C and C such that P p,n ( R v ≥ r ) ≤ P p c ,n ( R v ≥ r ) e − c ( p − p c ) r + C ( p − p c ) e − c ( p − p c ) r ≤ C (cid:18) r + ( p − p c ) (cid:19) e − c ( p − p c ) r ≤ C r e − c ( p − p c ) r/ for every 1 ≤ r, n < ∞ and p ∈ [ p c , p c + δ ). The claim follows by taking n → ∞ . The goal of the following two subsections is to prove the upper bound of Theorem 1.1 in the slightlysupercritical regime. This is the most technical part of the paper.
Proposition 3.7.
Let G = ( V, E ) be a connected, locally finite, quasi-transitive graph such that p c < p → . Then there exist positive constants δ , c , and C such that P p ( n ≤ | K v | < ∞ ) ≤ Cn − / exp h − c ( p − p c ) n i (3.13) for every n ≥ , v ∈ V , and p ∈ ( p c , p c + δ ) . To prove this proposition, it suffices to prove that there exist positive constants c , C , and δ such that E p, ∞ (cid:20) | K v | exp (cid:16) c ( p − p c ) | K v | (cid:17)(cid:21) ≤ Cp − p c (3.14)for every p ∈ ( p c , p c + δ ). Indeed, Markov’s inequality will then imply that P p ( n ≤ | K v | < ∞ ) ≤ C ( p − p c ) n exp h − c ( p − p c ) n i for every p ∈ ( p c , p c + δ ) and n ≥
1, which is of the correct order when n ≥ ( p − p c ) − . On theother hand, if n ≤ ( p − p c ) − then a bound of the correct order is already provided by Lemma 2.1.The primary remaining obstacle we must overcome in order to prove (3.14) is to establish upperbounds on U p,n [ | K v | k ], the positive part of the derivative of the truncated k th moment. Ourapproach will follow a similar philosophy to that of [29, Section 2.3]. Unfortunately, while themethods developed in that paper are quantitative, they are not sharp, and eventually lead to afactor of order ( p − p c ) rather than of order ( p − p c ) in the exponent of (3.13) when combined withour sharp control of skinny clusters, Proposition 3.5. Obtaining optimal bounds requires a rathermore delicate and technical approach. In this subsection, we derive a differential inequality whichwe will use to bound these quantities; the analysis of this differential inequality is then performedin the next subsection. We refer to this differential inequality as the auxiliary differential inequalityto distinguish it from the other differential inequalities we have been interested in.As in [29], we begin by expressing U p,n [ | K v | k ] geometrically in terms of bridges. We first recallthe relevant definitions. Let H be a connected graph. Recall that two vertices u and v of H are saidto be 2 -connected if u and v remain connected when any edge is deleted from H . (In particular,every vertex is 2-connected to itself.) Equivalently, by Menger’s theorem, u and v are 2-connected if18here exist a pair of edge-disjoint paths each connecting u to v . This defines an equivalence relationon the vertices of H , the pieces of which are referred to as the 2 -connected components of H .We write [ v ] for the 2-connected component of the vertex v in H . An edge e of H is said to be a bridge of H if the graph formed by deleting e from H is disconnected. Equivalently, e is a bridgeof H if its endpoints are in distinct 2-connected components of H . We define Tr( H ) to be the treewhose vertices are the 2-connected components of H and whose edges are the bridges of H . Givena graph H and a sequence of vertices v , . . . , v k of H , let Br( v , . . . , v k ; H ) be the number of edgesin the subtree of Tr( H ) spanned by the union of the geodesics between the vertices [ v ] , . . . , [ v k ] inthe tree of 2-connected components Tr( H ).Let G be a connected, locally finite, and quasi-transitive, let p ∈ [0 ,
1] and let v ∈ V . We haveby Proposition 3.1 that there exist positive constants c and δ such that if p c < p ≤ p c + δ then the p -derivative of E p,n h | K v | e u | K v | i satisfies ∂ p E p,n h | K v | e u | K v | i ≤ − c ( p − p c ) E p,n h | K v | e u | K v | i + U p,n h | K v | e u | K v | i = − c ( p − p c ) E p,n h | K v | e u | K v | i + ∞ X k =0 u k k ! U p,n h | K v | k +1 i (3.15)for every u ≥ n ≥
1. Observe that, by definition of the relevant quantities, we may express U p,n [ | K v | k ] as U p,n [ | K v | k ] = X x ,...,x k ∈ V ( G ) U p,n (cid:2) ( x , . . . , x k ∈ K v ) (cid:3) = 1 p X x ,...,x k ∈ V ( G ) E p,n (cid:2) ( x , . . . , x k ∈ K v ) Br( v, x , . . . , x k ; K v ) (cid:3) . Writing Br( v, x , . . . , x k ; K v ) = Br( v, x , . . . , x k ), this can be written more succinctly as U p,n [ | K v | k ] = 1 p E p,n X x ,...,x k ∈ K v Br( v, x , . . . , x k ) for every n, k ≥
1. Summing over k it follows that U p,n h | K v | e u | K v | i = 1 p ∞ X k =0 u k k ! E p,n X x ,...,x k ∈ K v Br( v, x , . . . , x k +1 ) (cid:18) Br( v, x , . . . , x k +1 ) ≥ cp ( p − p c ) | K v | (cid:19) + 1 p ∞ X k =0 u k k ! E p,n X x ,...,x k ∈ K v Br( v, x , . . . , x k +1 ) (cid:18) Br( v, x , . . . , x k +1 ) < cp ( p − p c ) | K v | (cid:19) n ≥ u ≥
0, from which we deduce that U p,n h | K v | e u | K v | i ≤ p ∞ X k =0 u k k ! E p,n X x ,...,x k ∈ K v Br( v, x , . . . , x k +1 ) (cid:18) Br( v, x , . . . , x k +1 ) ≥ cp ( p − p c ) | K v | (cid:19) + 12 c ( p − p c ) E p,n h | K v | e u | K v | i (3.16)for every u ≥ n ≥
1. Together, (3.15) and (3.16) imply that if G is a connected, locally finite,quasi-transitive graph with p c < p → then there exist constants δ , c , and c such that ∂ p E p,n h | K v | e u | K v | i ≤ − c ( p − p c ) E p,n h | K v | e u | K v | i + 1 p ∞ X k =1 u k k ! E p,n X x ,...,x k +1 ∈ K v Br( v, x , . . . , x k +1 ) (cid:16) Br( v, x , . . . , x k +1 ) ≥ c ( p − p c ) | K v | (cid:17) (3.17)for every v ∈ V , u ≥ p ∈ ( p c , p c + δ ) and 1 ≤ n < ∞ . Intuitively, the constraint thatBr( v, x , . . . , x k +1 ) ≥ c ( p − p c ) | K v | can be thought of as a higher-order version of the skinninessconstraint which we studied in Proposition 3.5.We will control the summands on the right hand side of (3.17) by an inductive analysis ofcertain generating functions, which we now introduce. Let G be a countable, locally finite graph,let p ∈ [0 , v be a vertex of G . For each k ≥ n ∈ N ∞ = { , , . . . } ∪ {∞} we define G k,n ( · , · ; G, v, p ) : R → [0 , ∞ ] by G k,n ( s, t ; G, v, p ) = ∞ X a =0 ∞ X b =0 X x ,...,x k ∈ V ( G ) P Gp,n (cid:16) x , . . . , x k ∈ K v , E v = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) e sa + tb , which is a sort of multivariate generating function, and also define F k,n ( s, t ; G, p ) := sup (cid:8) G k,n ( s, t ; H, u, p ) : H a subgraph of G , u a vertex of H (cid:9) . Finally, for each n ∈ N ∞ define M n ( · , · , · ; G, v, p ) : R × [0 , ∞ ) → [0 , ∞ ] by M n ( s, t, u ; G, p ) := ∞ X k =0 u k k ! F k +1 ,n ( s, t ; G, p ) . (3.18)Note that if G is a connected, locally finite, quasi-transitive graph with p c < p → and s ≤ c ( p − p c ) t then we have trivially that (cid:16) Br( v, x , . . . , x k +1 ; K v ) ≥ c ( p − p c ) | K v | (cid:17) ≤ exp( − s | K v | + t Br( v, x , . . . , x k +1 ; K v )) for every x , . . . , x k +1 ∈ K v and hence that the expression appearing on20he right hand side of (3.17) can be bounded1 p ∞ X k =1 u k k ! E p,n X x ,...,x k +1 ∈ K v Br( v, x , . . . , x k +1 ; K v ) (cid:16) Br( v, x , . . . , x k +1 ; K v ) ≥ c ( p − p c ) | K v | (cid:17) ≤ p ∞ X k =1 u k k ! E p,n X x ,...,x k +1 ∈ K v Br( v, x , . . . , x k +1 ; K v ) e − s | K v | + t Br( v,x ,...,x k +1 ; K v ) ≤ etp ∞ X k =1 u k k ! E p,n X x ,...,x k +1 ∈ K v e − s | K v | +2 t Br( v,x ,...,x k +1 ; K v ) ≤ etp M n ( − s, t, u ; G, p )for every u ≥ n ≥
1, where we used the elementary bound xe tx ≤ et − e tx in the secondinequality. It follows from this and (3.17) that if G is a connected, locally finite, quasi-transitivegraph with p c < p → then there exist positive constants δ , c , c , and C such that ∂ p E p,n h | K v | e u | K v | i ≤ − c ( p − p c ) E p,n h | K v | e u | K v | i + C t M n (cid:0) − c ( p − p c ) t, t, u ; G, p (cid:1) (3.19)for every v ∈ V , u ≥ p ∈ ( p c , p c + δ ), t ≥ ≤ n < ∞ .In order to apply the inequality (3.19), we will need to bound the generating function M n .To do this, we derive a family of recursive differential inequalities, Lemma 3.8, which in the nextsubsection we will use to bound the functions F k,n by an inductive argument.When n < ∞ all but finitely many terms of the sum defining G k,n ( s, t ; G, v, p ) are zero, so that G k,n ( s, t ; G, v, p ) is a differentiable function of ( s, t ) with t -derivative ∂ t G k,n ( s, t ; G, v, p ) = ∞ X a =0 ∞ X b =0 X x ,...,x k ∈ V ( G ) P Gp,n (cid:16) x , . . . , x k ∈ K v , E v = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) be sa + tb . = E p,n X x ,...,x k ∈ K v Br( v, x , . . . , x k ; K v ) e sE v + t Br( v,x ,...,x k ; K v ) . The following lemma can be thought of as a sharp form of [29, Lemma 2.10].
Lemma 3.8.
Let G be a countable graph with degrees bounded by M , let v be a vertex of G and let p ∈ (0 , . Then ∂ t G k,n ( s, t ; G, v, p ) ≤ M pe t − p k − X ℓ =0 (cid:18) kℓ (cid:19) G ℓ +1 ,n ( s, t ; G, v, p ) F k − ℓ,n ( s, t ; G, p ) for every k, n ≥ , and s, t ∈ R . Proof of Lemma 3.8.
Fix k, n ≥ p ∈ (0 , v ∈ V , and s, t ∈ R . For each a, b ≥ R k,n ( a, b ; G, v, p ) = X x ,...,x k ∈ V ( G ) b P Gp (cid:16) x , . . . , x k ∈ E v , | E v | = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) , so that ∂ t G k,n ( s, t ; G, v, p ) = ∞ X a =0 ∞ X b =0 e sa + tb R k,n ( a, b ; G, v, p ) . For each oriented edge e of G , let K − e and K + e be the connected components of e − and e + inthe subgraph of G spanned by the open edges of G other than e . Thus, K − e = K + e if and onlyif e − and e + are not connected to each other by an open path not containing e . Let E − e be thenumber of edges of G that touch K − e , and let E + e be the number of edges of G that touch K + e butdo not touch K − e . For each oriented edge e of G and each x , . . . , x k ∈ V , let A e ( x , . . . , x k ) be theevent that x , . . . , x k ∈ K v , that e is open, that v ∈ K − e , and that there exists 1 ≤ i ≤ k such that x i ∈ K + e \ K − e . For each x , . . . , x k ∈ K v , the number of oriented edges e such that A e ( x , . . . , x k )holds is precisely Br( v, x , . . . , x k ; K v ), so that we can write R k,n ( a, b ; G, v, p ) = X e ∈ E → X x ,...,x k ∈ V ( G ) P Gp,n (cid:16) A e ( x , . . . , x k ) , E v = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) . For each strict (possibly empty) subset A of { , . . . , k } , let B e ( x , . . . , x k ; A ) be the event in whichthe event that A e ( x , . . . , x k ) holds and that x i ∈ K − e if and only if i ∈ A for each 1 ≤ i ≤ k . Thenwe can expand R k,n ( a, b ; G, v, p ) = X e ∈ E → X x ,...,x k ∈ V ( G ) X A ⊂{ ,...,k } P Gp,n (cid:16) B e ( x , . . . , x k ; A ) , E v = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) , (3.20)and it follows by symmetry that R k,n ( a, b ; G, v, p ) = X e ∈ E → X x ,...,x k ∈ V ( G ) k − X ℓ =0 (cid:18) kℓ (cid:19) P Gp,n (cid:16) B e (cid:0) x , . . . , x k ; { , . . . , ℓ } (cid:1) , E v = a, Br( v, x , . . . , x k ; K v ) = b (cid:17) , (3.21)where we interpret { , . . . , ℓ } as the empty set when ℓ = 0.For each e ∈ E → , 0 ≤ ℓ ≤ k −
1, each y , . . . , y ℓ ∈ V ( G ), each z , . . . , z k − ℓ ∈ V ( G ), and each22 , a , b , b ≥
0, consider the events C e,ℓ ( y , . . . , y ℓ ; a , b ):= n v ∈ K − e , y i ∈ K − e for every 1 ≤ i ≤ ℓ , E − e = a , and Br( v, y , . . . , y ℓ , e − ; K − e ) = b o and D e,ℓ ( z , . . . , z k − ℓ ; a , b ):= n z i ∈ K + e \ K − e for every 1 ≤ i ≤ k − ℓ , E + e = a , and Br( e + , z , . . . , z k − ℓ ; K + e ) = b o . Observe that the event B e (cid:0) x , . . . , x k ; { , . . . , ℓ } (cid:1) ∩ { E v = a, | Br( v, x , . . . , x k ; K v ) | = b } can berewritten as the disjoint union B e (cid:0) x , . . . , x k ; { , . . . , ℓ } (cid:1) ∩ n E v = a, | Br( v, x , . . . , x k ; K v ) | = b o = a [ a =0 b − [ b =0 (cid:20) { e open } ∩ C e,ℓ (cid:16) x , . . . , x ℓ ; a , b (cid:17) ∩ D e,ℓ (cid:16) x ℓ +1 , . . . , x k ; a − a , b − b − (cid:17)(cid:21) . (3.22)Indeed, this follows from the observation that if B e (cid:0) x , . . . , x k ; { , . . . , ℓ } (cid:1) holds then E v = E − e + E + e and Br( v, x , . . . , x k ; K v ) = Br( v, x , . . . , x ℓ , e − ; K − e ) + Br( e + , x ℓ +1 , . . . , x k ; K + e ) + 1. Noting thatthe random variable ω ( e ) is independent of the pair of random variables ( K − e , K + e ), we deduce from(3.21) and (3.22) that R k,n ( a, b ; G, v, p ) = p k − X ℓ =0 (cid:18) kℓ (cid:19) X e ∈ E → a X a =0 b − X b =0 X y ,...,y ℓ ∈ V ( G ) X z ,...,z k − ℓ ∈ V ( G ) P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1) ∩ D e,ℓ (cid:0) z , . . . , z k − ℓ ; a − a , b − b − (cid:1)(cid:17) and hence that ∂ t G k,n ( s, t ; G, v, p ) = pe t k − X ℓ =0 (cid:18) kℓ (cid:19) X e ∈ E → ∞ X a =0 ∞ X b =0 X y ,...,y ℓ ∈ V ( G ) ∞ X a =0 ∞ X b =0 X z ,...,z k − ℓ ∈ V ( G ) e sa + tb e sa + tb P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1) ∩ D e,ℓ (cid:0) z , . . . , z k − ℓ ; a , b (cid:1)(cid:17) . (3.23)Let F − e be the σ -algebra generated by the random variable K − e and let H + e be the randomsubgraph of G spanned by those edges of G that do not touch K − e . The conditional distributionof K + e \ K − e given F − e coincides with that of the cluster of e + in Bernoulli- p bond percolation on23 + e , so that ∞ X a =0 ∞ X b =0 X z ,...,z k − ℓ ∈ V ( G ) P Gp,n (cid:16) D e,ℓ (cid:0) z , . . . , z k − ℓ ; a , b (cid:1) | F − e (cid:17) e sa + tb = X z ,...,z k − ℓ ∈ V ( H + e ) P H + e p,n − E − v (cid:16) z , . . . , z k − ℓ ∈ K e + , E e + = a , Br( e + , z , . . . , z k − ℓ ; K e + ) = b (cid:17) e sa + tb = G k − ℓ,n − E − v ( s, t ; H + e , e + , p ) ≤ F k − ℓ,n − E − v ( s, t ; G, p ) ≤ F k − ℓ,n ( s, t ; G, p )almost surely. Since C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1) is F − e -measurable, it follows that ∂ t G k,n ( s, t ; G, v, p ) ≤ pe t k − X ℓ =0 (cid:18) kℓ (cid:19) F k − ℓ,n ( s, t ; G, p ) X e ∈ E → ∞ X a =0 ∞ X b =0 X y ,...,y ℓ ∈ V ( G ) P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1)(cid:17) e sa + tb . (3.24)On the other hand, since ω ( e ) is independent of F − e we have that P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1)(cid:17) = 11 − p P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1) ∩ { e closed } (cid:17) = 11 − p P Gp,n (cid:16) e closed, y , . . . , y ℓ , e − ∈ K v , E v = a , and Br( v, y , . . . , y ℓ , e − ; K v ) = b (cid:17) ≤ − p P Gp,n (cid:16) y , . . . , y ℓ , e − ∈ K v , E v = a , and Br( v, y , . . . , y ℓ , e − ; K v ) = b (cid:17) and hence that X e ∈ E → ∞ X a =0 ∞ X b =0 X y ,...,y ℓ ∈ V ( G ) P Gp,n (cid:16) C e,ℓ (cid:0) y , . . . , y ℓ ; a , b (cid:1)(cid:17) e sa + tb ≤ M − p ∞ X a =0 ∞ X b =0 X y ,...,y ℓ +1 ∈ V ( G ) P Gp,n (cid:0) y , . . . , y ℓ , y ℓ +1 ∈ K v , E v = a , Br( v, y , . . . , y ℓ +1 ; K v ) = b (cid:1) e sa + tb = M − p G ℓ +1 ,n ( s, t ; G, v, p ) . (3.25)Substituting (3.25) into (3.24) completes the proof. In this subsection we complete the proof of Proposition 3.7. The main step will be to prove thefollowing proposition via an analysis of the recursive differential inequality provided by Lemma 3.8.24his proposition serves as a sharp quantitative version of [29, Proposition 2.7] under the additionalassumption that ∇ p c < ∞ . The generating function M n was defined in (3.18). Proposition 3.9.
Let G be an infinite, connected, locally finite, quasi-transitive graph such that ∇ p c < ∞ , and let α ≥ . Then there exist positive constants c = c ( G, α ) , c = c ( G, α ) , C = C ( G, α ) , and δ = δ ( G, α ) such that M n ( − c ε , αc ε, c ε ; G, p c + ε ) ≤ Cε − for every n ≥ and < ε ≤ δ . Let G be a countable graph with degrees bounded by M , let v be a vertex of G and let p ∈ (0 , F k,n ( s, t ; H, p ) ≤ F k,n ( s, t ; G, p ) for every subgraph H of G , integrating thedifferential inequality provided by Lemma 3.8 and then taking suprema over subgraphs yields that F k,n ( s, t ; G, p ) − F k,n ( s, t ; G, p ) ≤ M p − p k − X ℓ =0 (cid:18) kℓ (cid:19) Z t t = t e t F ℓ +1 ,n ( s, t ; G, p ) F k − ℓ,n ( s, t ; G, p ) d t (3.26)for every k, n ≥ s, t , t ∈ R with t ≤ t . We will prove Proposition 3.9 by an inductiveanalysis of this integral inequality. This analysis will require the following two lemmas as input:The first applies Lemma 2.1 to analyze F k, ∞ when t = 0 and s <
0, and the second appliesProposition 3.5 to establish the k = 1 base case. Lemma 3.10.
Let G be an infinite, connected, locally finite quasi-transitive graph such that ∇ p c < ∞ . Then there exist positive constants C , and δ such that F k, ∞ (cid:0) − λε , G, p c + ε (cid:1) ≤ k ! C k λ − k ε − k +1 (3.27) for every k ≥ , < ε ≤ δ , and < λ ≤ . Lemma 3.11.
Let G be an infinite, connected, locally finite, quasi-transitive graph such that ∇ p c < ∞ . Then there exist positive constants c , C , and δ such that F , ∞ (cid:0) − λε , αλε ; G, p c + ε (cid:1) ≤ Cλ − ε − (3.28) for every α ≥ , < ε ≤ δ , and < λ ≤ ∧ cα − e − Cα . The proofs of both lemmas will use the fact that if X is a non-negative random variable then E h X k e sX i = Z ∞ t =0 ( k + st ) t k − e st P ( X ≥ t ) d t (3.29)for every k ≥ s ∈ R , where it is possible that both sides are equal to + ∞ when s ≥
0. Thisidentity is a standard consequence of the integration-by-parts formula.
Proof of Lemma 3.10.
Let p c = p c ( G ). We have by Lemma 2.1 that there exist positive constants C and δ such that P Hp c + ε ( E v ≥ n ) ≤ C h n − / + ε i H of G , every vertex v of H , every n ≥ < ε ≤ δ . Since | K v | ≤ E v , we deduce by standard calculations that G k, ∞ ( − s, H, v, p c + ε ) = ∞ X a =0 X x ,...,x k ∈ V ( G ) P Gp (cid:16) x , . . . , x k ∈ K v , E v = a (cid:17) e − sa = E p c + ε h | K v | k e − sE v i ≤ E Hp c + ε h (1 + E v ) k e − sE v i ≤ Z ∞ ( k − su ) u k − e − su P Hp ( E v ≥ u ) d u ≤ C (cid:20) Z ∞ ku k − / e − su d u + ε Z ∞ ku k − e − su d u (cid:21) for every s >
0, where we used that (1 + x ) k ≤ x k ) for every x ≥ k ≥ R ∞ u a − e − su d u = s − a R ∞ y a − e − y d y = s − a Γ( a ) and Γ( k ) = ( k − G k, ∞ ( − s, H, v, p c + ε ) ≤ C h ks − k +1 / Γ( k − /
2) + kεs − k Γ( k ) i ≤ C k ! h s − k +1 / + εs − k i for every subgraph H of G , every vertex v of H , every n ≥
0, every 0 < ε ≤ δ , and every s ≥ s = λε . Proof of Lemma 3.11.
Fix α ≥
0, a subgraph H of G and a vertex v of H . Letting R v denote theintrinsic radius of the cluster of v in H , we trivially have that Br( v, x ; K v ) ≤ R v for every x ∈ K v ,so that G , ∞ ( − λε , αλε ; H, v, p c + ε ) = E Hp c + ε e − λε E v X x ∈ K v e αλε Br( v,x ; K v ) ≤ E Hp c + ε h | K v | e − λε E v e αλεR v i and hence that G , ∞ ( − λε , αλε ; H, v, p c + ε ) ≤ E Hp c + ε " | K v | e − λε E v e αλεR v (cid:18) R v ≤ εE v α (cid:19) + E Hp c + ε " | K v | e − λε E v e αλεR v (cid:18) R v > εE v α (cid:19) ≤ E Hp c + ε h | K v | e − λε E v / i + E Hp c + ε " | K v | e αλεR v (cid:18) R v > εE v α (cid:19) (3.30)for every 0 < ε ≤ δ and 0 < λ ≤
1. For the first term, Lemma 3.10 implies that there exist positiveconstants δ and C such that E Hp c + ε h | K v | e − λε E v / i ≤ F k, ∞ (cid:16) − λε / , G, p c + ε (cid:17) ≤ C λ − ε − (3.31)for every 0 < ε ≤ δ and 0 < λ ≤
1. 26or the second term, we first decompose further E Hp c + ε " | K v | e αλεR v (cid:18) R v > εE v α (cid:19) = E Hp c + ε " | K v | e αλεR v (cid:18) R v > εE v α , R v < ε − (cid:19) + E Hp c + ε " | K v | e αλεR v (cid:18) R v > εE v α , R v ≥ ε − (cid:19) =: I + II , where the second inequality means that we write I and II for the first and second terms appearingon the right hand side of the first equality. To bound the term I, we apply (2.2) to deduce thatthere exist positive constants δ and C such thatI ≤ e αλ E Hp c + ε h | K v | ( R v < ε − ) i ≤ e αλ E Hp c + ε h B int ( v, ⌊ ε − ⌋ ) i ≤ C ε − e αλ , (3.32)for every 0 ≤ ε ≤ δ and 0 < λ ≤
1. Finally, to bound the term II, we note thatII ≤ αε E Hp c + ε " R v e αλεR v (cid:18) R v > εE v α , R v ≥ ε − (cid:19) (where we used the fact that | K v | ≤ E v + 1 ≤ E v when R v ≥ ε − >
0) and hence by (3.29) thatthere exists a constant C such thatII ≤ C αε Z ∞ ε − (1 + αλεt ) e αλεt P Hp c + ε (cid:18) t ≤ R v < ∞ , E v < αε R v (cid:19) d t. We then apply Proposition 3.5 to obtain that there exist positive constants c , C , C and δ suchthat II ≤ C αε Z ∞ t = ε − ( ε + αλε t ) exp h − c e − C α εt + αλεt i d t for every 0 < ε ≤ δ and 0 < λ ≤
1. We deduce in particular that if 0 < ε ≤ δ and αλ ≤ c e − C α then II ≤ C αε Z ∞ t = ε − (cid:0) ε + αλε t ) exp h − c e − C α εt i d t. Similarly to the proof of Lemma 3.10, using the identities R ∞ e − st d t = s − and R ∞ te − st d t = s − yields that there exists a constant C such thatII ≤ C αε h e C α + αλe C α i ≤ C α (1 + αλ ) e C α ε − (3.33)for every 0 < ε ≤ δ and 0 < λ ≤ ∧ c α − e − C α .Putting together all the estimates (3.30), (3.31), (3.32), and (3.33), we obtain that there existpositive constants δ = δ ∧ δ ∧ δ and C such that G , ∞ ( − λε , αλε ; H, v, p c + ε ) ≤ C h λ − + e αλ + α (1 + αλ ) e C α i ε − < ε ≤ δ and 0 < λ ≤ ∨ c α − e − C α . It follows that there exists a constant C suchthat if 0 < ε ≤ δ and 0 < λ ≤ ∧ c α − e − C α ∧ e − α then G , ∞ ( − λε , αλε ; H, v, p c + ε ) ≤ C λ − ε − . Since H , v , and α ≥ Proof of Proposition 3.9.
Let G be an infinite, connected, locally finite, quasi-transitive graph, let p c = p c ( G ), and suppose that ∇ p c < ∞ . Let α ≥
0. It suffices to prove there exist positiveconstants c = c ( G, α ), C = C ( G, α ), and δ = δ ( G, α ) such that F k, ∞ (cid:16) − cε , αcε ; G, p c + ε (cid:17) ≤ ( k − C k ε − k +1 (3.34)for every k ≥ < ε ≤ δ . Indeed, the claim will then follow by noting that if (3.34) holdsthen M ∞ (cid:0) − cε , αcε, C ε ; G, p c ( G ) + ε (cid:1) ≤ ∞ X k =0 ε k k !2 k C k k ! C k +1 ε − k +1)+1 = 2 Cε − for every 0 < ε ≤ δ . The case α = 0 is handled by Lemma 3.10, so we may suppose that α > α and G . Throughout the proofwe will also use the convention that ( − C such that1( k − k − X ℓ =1 (cid:18) kℓ (cid:19) ( ℓ − k − ℓ − k − X ℓ =1 kℓ ( k − ℓ )( k − ℓ −
1) + kk − ≤ k − ⌊ ( k − / ⌋ X ℓ =1 ℓ + k − X ℓ = ⌈ ( k − / ⌉ k ( k − k − ℓ )( k − ℓ −
1) + kk − ≤ C for every k ≥
2. By Lemma 3.11, there exist positive constants c , C , and δ such that F , ∞ ( − λε , αλε ; G, p c + ε ) ≤ C ε − (3.35)for every 0 ≤ λ ≤ c and 0 < ε ≤ δ . Define c = min (cid:26) , c , − p c αC eM , − p c αC eM (cid:27) , where M is the maximum degree of G . For each k, n ≥ < ε ≤ δ we define increasingfunctions f k,n,ε : [0 , → [0 , ∞ ) and f k,ε : [0 , → [0 , ∞ ] by f k,n,ε ( θ ) = F k,n ( − cε , αcεθ ; G, p c ( G ) + ε ) and f k,ε ( θ ) = F k, ∞ ( − cε , αcεθ ; G, p c ( G ) + ε ) , so that f k,ε ( θ ) = sup n ≥ f k,n,ε ( θ ). Thus, (3.35) implies that f ,ε ( θ ) ≤ f ,ε (1) ≤ C ε − for every ε ≤ δ and θ ∈ [0 , C δ , such that f k,ε (0) ≤ C k ε − k +1 ( k − < ε ≤ δ and k ≥ δ = min { δ , δ , (1 − p c ) / } and let C = 4( C ∨ C ) >
0. It suffices to prove that f k,ε ( θ ) ≤ C k e C ( k − θ ε − k +1 ( k − k, n ≥ θ ∈ [0 ,
1] and 0 < ε ≤ δ . We will do this by induction on k . The base case k = 1follows immediately from (3.35). Suppose that k ≥ ≤ k ′ < k .Fix n ≥ < ε ≤ δ . It follows from eq. (3.26) that f k,n,ε ( θ ) ≤ f k,n,ε (0) + M ( p c + ε )1 − ( p c + ε ) k − X ℓ =0 (cid:18) kℓ (cid:19) Z θϕ =0 e cεϕ f ℓ +1 ,n,ε ( ϕ ) f k − ℓ,n,ε ( ϕ ) αcε d ϕ, where the αcε term comes from changing variables in the integral from t to ϕ . Our choice of c therefore yields that f k,n,ε ( θ ) ≤ f k,ε (0) + eM αcε − p c − δ Z θϕ =0 f k,n,ε ( ϕ ) f ,ε ( ϕ ) d ϕ + eM αcε − p c − δ k − X ℓ =1 (cid:18) kℓ (cid:19) Z θϕ =0 f ℓ +1 ,ε ( ϕ ) f k − ℓ,ε ( ϕ ) d ϕ ≤ f k,ε (0) + ε C Z θϕ =0 f k,n,ε ( ϕ ) f ,ε ( ϕ ) d ϕ + ε C k − X ℓ =1 (cid:18) kℓ (cid:19) Z θϕ =0 f ℓ +1 ,ε ( ϕ ) f k − ℓ,ε ( ϕ ) d ϕ (3.38)for every θ ∈ [0 , e cεϕ ≤ e c ≤ e in the first line. The first two terms are easilybounded by using (3.35), (3.36), and the definition of C to obtain that f k,ε (0) + ε C Z θϕ =0 f k,n,ε ( ϕ ) f ,ε ( ϕ ) d ϕ ≤ C k ε − k +1 ( k − ε C Z θϕ =0 f k,n,ε ( θ ) C ε − d ϕ ≤ C k ε − k +1 ( k − f k,n,ε ( θ ) (3.39)for every θ ∈ [0 , C to obtain that ε C k − X ℓ =1 (cid:18) kℓ (cid:19) Z θϕ =0 f ℓ +1 ,ε ( ϕ ) f k − ℓ,ε ( ϕ ) d ϕ ≤ ε C k − X ℓ =1 (cid:18) kℓ (cid:19) Z θϕ =0 C ℓ +14 e C ℓϕ ε − ℓ − ( ℓ − C k − ℓ e C ( k − ℓ − ϕ ε − k − ℓ +1 ( k − ℓ − ϕ = C k +14 ε − k +1 C k − X ℓ =1 (cid:18) kℓ (cid:19) ( ℓ − k − ℓ − Z θϕ =0 e C ( k − ϕ d ϕ = C k ε − k +1 C k − X ℓ =1 (cid:18) kℓ (cid:19) ( ℓ − k − ℓ − k − e C ( k − θ ≤ C k e C ( k − θ ε − k +1 ( k − θ ∈ [0 , f k,n,ε ( θ ) ≤ C k ε − k +1 ( k − f k,n,ε ( θ ) + 14 C k e C ( k − θ ε − k +1 ( k − θ ∈ [0 , f k,n,ε ( θ ) ≤ C k e C ( k − θ ε − k +1 ( k − θ ∈ [0 ,
1] as desired. Since n ≥ < ε ≤ δ were arbitrary, taking n → ∞ completesthe induction step and hence also the proof. Remark . The correct form of the induction hypothesis (3.37) needed to make this argumentwork was not at all obvious to us, and was found by extensive trial and error. We would beinterested to know if someone is aware of a more systematic way of approaching similar problems.It remains to deduce Proposition 4.1 from Proposition 3.9.
Proof of Proposition 3.7.
Fix v ∈ V . By (3.19) there exist positive constants c , c , δ , and C such that ∂ p E p,n h | K v | e u | K v | i ≤ − c ( p − p c ) E p,n h | K v | e u | K v | i + C t M n (cid:0) − c ( p − p c ) t, t, u ; G, p (cid:1) = − c ( p − p c ) ∂ u E p,n h | K v | e u | K v | i + C t M n (cid:0) − c ( p − p c ) t, t, u ; G, p (cid:1) for every u ≥ p ∈ ( p c , p c + δ ), t ≥ ≤ n < ∞ . On the other hand, applying Proposition 3.9with α = c − yields that there exist positive constants c , c , δ , and C such that M n (cid:0) − c c ( p − p c ) , c ( p − p c ) , u ; G, p (cid:1) ≤ M n (cid:0) − c c ( p − p c ) , c ( p − p c ) , c ( p − p c ) ; G, p (cid:1) ≤ C ( p − p c ) − for every p ∈ ( p c , p c + δ ) and 0 ≤ u ≤ c ( p − p c ) . It follows that there exists a constant C suchthat ∂ p E p,n h | K v | e u | K v | i ≤ − c ( p − p c ) ∂ u E p,n h | K v | e u | K v | i + C ( p − p c ) − (3.41)for every p ∈ ( p c , p c + δ ∧ δ ) and 0 ≤ u ≤ c ( p − p c ) .By Corollary 3.3, there exists δ > C < ∞ such that for every 0 < ε ≤ δ there exists p = p ( ε ) ∈ ( p c + ε/ , p c + ε/
2) such that E p , ∞ | K v | ≤ C ε − . Let δ = min { δ , δ , δ } and c = ( c ∧ c ) /
2. Let 0 < ε ≤ δ and let p = p ( ε ). It follows by the chain rule that ddp E p,n (cid:20) | K v | exp h c ( p − p ) | K v | i(cid:21) ≤ C ( p − p c ) − for every n ≥ p ≤ p ≤ p c + ε . Integrating this differential inequality between p and p c + ε ε/ ≤ p c + ε − p ≤ ε yields that E p c + ε,n " | K v | exp (cid:20) c ε | K v | (cid:21) ≤ E p c + ε,n (cid:20) | K v | exp h c ( p c + ε − p ) | K v | i(cid:21) = E p (cid:2) | K v | (cid:3) + Z p c + εp ddp E p,n (cid:20) | K v | exp h c ( p − p ) | K v | i(cid:21) d p ≤ C ε − + Z p c + εp C ( p − p c ) − d p ≤ ( C + 16 C ) ε − for every n ≥ < ε ≤ δ as required. Remark . Consider the generating function N n ( s, t, u ) = N n ( s, t, u ; G, p ) = ∞ X k =1 u k k ! F k,n ( s, t ; G, p ) , which satisfies ∂ u N n = M n . Summing the differential inequality given by Lemma 3.8 over k ≥ ∂ t N n ≤ M pe t − p N n ∂ u N n ∀ s, t ∈ R , u ≥ , n ≥ . (3.42)(Note that while N n need not be differentiable, it is locally Lipschitz and hence differentiable almosteverywhere.) See e.g. the discussion of exponential generating functions in [60]. This point of viewmay be a useful starting point for further analysis. (It appeared to us to be ill-suited to our presentaims, however.) In this section we complete the proof of Theorems 1.1 and 1.2. It remains to establish lower boundsin the slightly supercritical regime, as well as both upper and lower bounds in the critical andslightly subcritical regimes. Several of these bounds are closely related to estimates that havealready been proven in the literature, but still require a delicate treatment to establish in thedesired sharp form.We begin by proving upper bounds in the critical and slightly subcritical regimes under theassumption that ∇ p c < ∞ . Proposition 4.1 (Subcritical upper bounds) . Let G be an infinite, connected, locally finite, quasi-transitive graph, and suppose that ∇ p c < ∞ . Then there exists positive constants c and C suchthat P p (cid:0) Rad ext ( K v ) ≥ r (cid:1) ≤ P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≤ Cr exp (cid:0) − c | p − p c | r (cid:1) (4.1) and P p (cid:0) | K v | ≥ n (cid:1) ≤ Cn / exp (cid:16) − c | p − p c | n (cid:17) (4.2) for every n, r ≥ , ≤ p ≤ p c , and v ∈ V . ≍ , (cid:22) , and (cid:23) for equalities and inequalities that hold to within multiplicationby a positive constant depending only on G . Proof.
Fix v ∈ V and write R v = Rad int ( K v ). As discussed in the introduction, it is known that if G is an infinite, connected, locally finite, quasi-transitive graph with ∇ p c < ∞ , then P p c ( | K v | ≥ n ) ≍ n − / and P p c ( R v ≥ r ) ≍ r − (4.3)for every n, r ≥
1, and E p (cid:2) | K v | (cid:3) ≍ ( p − p c ) − (4.4)for every 0 ≤ p ≤ p c . These results essentially follow from the works of Barsky and Aizenman [6],Kozma and Nachmias [40], and Aizenman and Newman [5]. These papers all dealt with the case G = Z d , see [33, Section 7] for a discussion of how to generalize these results to arbitrary quasi-transitive graphs with ∇ p c < ∞ . It follows from (4.4) and the tree-graph method of Aizenman andNewman [5] (see also [25, Chapter 6.3]) that there exists a constant C such that E p h | K v | k i ≤ k ! C k | p − p c | − k +1 for every k ≥ p < p c and hence that there exists a constant c = 1 / C such that E p h | K v | e c | p − p c | | K v | i ≤ ∞ X k =0 | p − p c | k k C k k ! ( k + 1)! C k +11 | p − p c | − k − = ∞ X k =0 C k − k | p − p c | − (cid:22) | p − p c | − for every 0 ≤ p < p c . Markov’s inequality then implies that P p (cid:0) | K v | ≥ n (cid:1) (cid:22) n ( p − p c ) exp h − c | p − p c | n i (4.5)for every 0 ≤ p < p c , and together with (4.3) this implies the desired bound (4.2). (Indeed, simplyuse the bound (4.5) if n ≥ ( p − p c ) − and the bound (4.3) otherwise, noting that P p ( | K v | ≥ n ) isincreasing in p .) See also [35] for an alternative derivation of the inequality (4.5) from (4.3).It remains to prove (4.1). The case r ≤ | p − p c | − is already handled by Lemma 2.1, so it sufficesto consider the case r ≥ | p − p c | − . We have by the union bound that P p ( R v ≥ r ) ≤ P p (cid:16) | K v | ≥ | p − p c | − r (cid:17) + P p (cid:16) R v ≥ r and | K v | ≤ | p − p c | − r (cid:17) . Using (4.2) to bound the first term and Proposition 3.5 with λ = | p − p c | to bound the secondyields that there exist positive constant c such that P p ( R v ≥ r ) (cid:22) r exp (cid:2) − c | p − p c | r (cid:3) + (cid:18) r + | p − p c | (cid:19) exp (cid:2) − c | p − p c | r (cid:3) , which is easily seen to be of the required order (since xe − xr ≤ er − e − xr/ for every x ∈ R ).We next study the intrinsic radius in the subcritical case. This is our only bound that holds for all quasi-transitive graphs. 32 roposition 4.2. Let G be an infinite, connected, locally finite, quasi-transitive graph. Then thereexist positive constants c , C , and δ such that P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ cr exp (cid:0) − C | p − p c | r (cid:1) for every p ∈ ( p c − δ, p c ] , r ≥ , and v ∈ V .Proof. A similar argument to that of Lemma 2.1 establishes that P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ P q (cid:0) Rad int ( K v ) ≥ r (cid:1) exp (cid:20) − q − pp r (cid:21) for every 0 ≤ p ≤ q ≤ r ≥
1. It follows in particular that P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ exp (cid:20) − (1 − p ) p r (cid:21) for every 0 ≤ p ≤ r ≥
1, which implies the claim in the case p c = 1. On the other hand,if p c < c such that P p ( v → ∞ ) ≥ c ( p − p c ) for every p c ≤ p ≤ P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ c ( q − p c ) exp (cid:20) − q − pp r (cid:21) for every 0 ≤ p ≤ p c ≤ q ≤
1. Taking q − p c = ( p c − p ) ∧ r − implies the claim.We next prove sharp lower bounds on the tail of the volume under the assumption that ∇ p c < ∞ . Proposition 4.3.
Let G be an infinite, connected, locally finite, quasi-transitive graph, and supposethat ∇ p c < ∞ . Then there exist positive constants c , C , and δ such that P p (cid:0) n ≤ | K v | < ∞ (cid:1) ≥ cn − / exp (cid:16) − C | p − p c | n (cid:17) (4.6) for every p ∈ ( p c − δ, p c + δ ) , r ≥ , and v ∈ V .Remark . Nguyen [50] proved that, under the same conditions as Proposition 4.3, there existconstants ( C k ) k ≥ such that E p h | K v | k i ≥ C k | p − p c | − k +1 for every 0 ≤ p < p c and k ≥
1. This is sufficient to determine the value of the gap exponent∆ = 2. However, it seems that the argument of [50] does not give sharp ( C k ≥ k ! e − O ( k ) ) controlof the value of the constant C k , and therefore does not establish the subcritical case of the bound(4.6). Similarly, classical arguments of Durrett and Nguyen [21] and Newman [49] can be used toprove related inequalities for the truncated k th moment E p, ∞ h | K v | k i in the slightly supercriticalregime. Again, however, it appears that these estimates are not sharp, and lose various logarithmicfactors compared to our estimate (4.6). Proof of Proposition 4.3.
Write R v = Rad int ( K v ). First suppose that p ≤ p c . Taking λ = α | p − p c |
33n Proposition 3.5, we obtain that that there exist positive constants c and C such that P p (cid:0) | K v | ≤ n, R v ≥ α | p − p c | n (cid:1) ≤ C (cid:18) α | p − p c | n + α | p − p c | (cid:19) exp h − c α | p − p c | n i for every 0 ≤ p ≤ p c , n ≥
1, and α ≥
1. Letting c , C , and δ be the constants from Proposition 4.2,it follows that P p (cid:0) | K v | ≥ n (cid:1) ≥ P p (cid:0) R v ≥ α | p − p c | n (cid:1) − P p (cid:0) | K v | ≤ n, R v ≥ α | p − p c | n (cid:1) ≥ c α | p − p c | n exp h − C α | p − p c | n i − C (cid:18) α | p − p c | n + α | p − p c | (cid:19) exp h − c α | p − p c | n i for every p c − δ ≤ p ≤ p c , r ≥
1, and α ≥
1. Taking α = 1 ∨ (2 C /c ) we deduce that there existpositive constants c , C , and C such that P p (cid:0) | K v | ≥ n (cid:1) ≥ c | p − p c | n exp h − C | p − p c | n i − C (cid:18) | p − p c | n + | p − p c | (cid:19) exp h − C | p − p c | n i for every p ∈ ( p c − δ, p c ) and n ≥
1. It follows readily that there exist positive constants c and C such that P p (cid:0) | K v | ≥ n (cid:1) ≥ c √ n exp h − C | p − p c | n i for every p ∈ ( p c − δ, p c ) and n ≥ C | p − p c | − . Since P p ( | K v | ≥ n ) is decreasing in n , it followsthat P p (cid:0) | K v | ≥ n (cid:1) ≥ c p n ∨ C | p − p c | − exp h − C | p − p c | ( n ∨ C | p − p c | − ) i (4.7)for every p ∈ ( p c − δ, p c ) and n ≥ p ≤ p c and n if of order at most | p − p c | − . It follows from theproof of [34, Proposition 3.6] thatsup u ∈ V E p c r X ℓ = r ∂B int ( u, ℓ ) ≥ r + 1for every r ≥
1, and an argument similar to that performed in the proof of Lemma 2.1 shows thatthere exist constants δ ≤ p c / C such thatsup u ∈ V E p r X ℓ = r ∂B int ( u, ℓ ) ≥ ( r + 1) exp (cid:20) − p − p c ) p r (cid:21) ≥ ( r + 1) exp (cid:2) − C | p − p c | r (cid:3) (4.8)34or every p ∈ ( p c − δ , p c ] and r ≥
1. Applying (4.3), it follows thatsup u ∈ V E p r X ℓ = r ∂B int ( u, ℓ ) | R u ≥ r ≥ c ( r + 1) exp (cid:2) − C | p − p c | r (cid:3) (4.9)for every p ∈ ( p c − δ , p c ] and r ≥
1. On the other hand, since ∇ p c < ∞ , it is known [40, 53] thatthere exists a constant C such that E p (cid:2) B int ( u, r ) (cid:3) ≤ C ( r + 1)for every u ∈ V , 0 ≤ p ≤ p c and r ≥
0. A straightforward and well-known variation on thetree-graph inequality method of Aizenman and Newman [5] gives that E p h ( B int ( u, r )) i ≤ sup w ∈ V E p (cid:2) ( B int ( w, r )) (cid:3) for every u ∈ V and r ≥
1, and hence that E p h ( B int ( u, r )) i ≤ C ( r + 1) for every r ≥ u ∈ V and 0 ≤ p ≤ p c . It follows from the Paley-Zygmund inequality that P p | K u | ≥ E p r X ℓ = r ∂B int ( u, ℓ ) | R u ≥ r ≥ E p hP rℓ = r ∂B int ( u, ℓ ) i E p (cid:2) ( B int ( u, r )) (cid:3) (4.10)for every u ∈ V , 0 < p ≤ p c , and r ≥
1. Applying this inequality together with (4.8), (4.9), and(4.10) and maximizing over u , it follows thatsup u ∈ V P p (cid:18) | K v | ≥ c r + 1) e − C | p − p c | r (cid:19) ≥ e − C | p − p c | r C ( r + 1)for every p ∈ ( p c − δ , p c ] and r ≥
1. Since G is connected and quasi-transitive, it follows straight-forwardly that there exist constants c , c , and c such that P p (cid:0) | K v | ≥ n (cid:1) ≥ c sup u ∈ V P p (cid:0) | K u | ≥ n (cid:1) ≥ c √ n for every p ∈ ( p c − δ , p c ] and 1 ≤ n ≤ c | p − p c | − . The claimed bound (4.6) follows in the case p ∈ ( p c − δ , p c ] from this together with (4.7).We now consider the case p ≥ p c . Let ω p and ω p c be Bernoulli- p and Bernoulli- p c percolationon G coupled in the standard monotone way, so that, conditional on ω p c , every ω p c -open edge is ω p open and every ω p c -closed edge is chosen to be either ω p -open or ω p -closed independently atrandom with probability ( p − p c ) / (1 − p c ) = O ( p − p c ) to be ω p -open. Let K p c v and K pv denote the35lusters of v in ω p c and ω p respectively. By (4.3), there exist constants c and C such that P ( n ≤ | K p c v | ≤ αn ) ≥ c √ n − C √ αn for every n ≥ α ≥
1. Taking α = C := 1 ∨ (2 C /c ) , it follows that there exists a positiveconstant c such that P ( n ≤ | K p c v | ≤ C n ) ≥ c √ n (4.11)for every n ≥
1. Let A n be the event that n ≤ | K p c v | ≤ C n and let B n be the event that n ≤ | K pv | < ∞ . If A n occurs but B n does not, then there must exist an ω p c -closed edge in theboundary of K p c v that is ω p -open and whose other endpoint is connected to infinity in ω p by an openpath that does not visit any vertex of K p c v . Conditional on K p c v , the probability that any particularedge in the boundary of K p c v has this property is bounded by ( p − p c ) θ ∗ ( p ) / (1 − p c ) = O (( p − p c ) ),and it follows by the FKG inequality that there exists a constant C such that P ( B n | K p c v ) ≥ ( A n ) (cid:20) − ∧ ( p − p c ) θ ∗ ( p )1 − p c (cid:21) M | K pcv | ≥ ( n ≤ | K p c v | ≤ C n ) e − C ( p − p c ) n , (4.12)where M is the maximum degree of G . The claimed bound follows from (4.11) and (4.12) by takingexpectations over K p c v .Finally, we prove a lower bound on the tail of the radius of a finite cluster in the supercriticalregime under the assumption that p c < p → . Proposition 4.5.
Let G be an infinite, connected, locally finite, quasi-transitive graph, and supposethat p c < p → . Then there exist positive constants c and C such that P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ P p (cid:0) Rad ext ( K v ) ≥ r (cid:1) ≥ cr exp (cid:0) − C | p − p c | r (cid:1) (4.13) for every r ≥ and p ∈ ( p c − δ, p c + δ ) .Proof of Proposition 4.5. By Proposition 4.2 there exist positive constants c , C , and δ such that P p (cid:0) Rad int ( K v ) ≥ r (cid:1) ≥ c r exp (cid:0) − C | p − p c | r (cid:1) (4.14)for every r ≥ p ∈ ( p c − δ , p c ). On the other hand, it follows from [34, Proposition 3.2] that P p (cid:0) Rad int ( K v ) ≥ r and Rad ext ( K v ) ≤ ℓ (cid:1) ≤ k T p k → exp " − re k T p k → | B ( v, ℓ ) | / for every 0 ≤ p < p → and r, ℓ ≥
1. Since p c < p → , it follows that there exist constants c , c , C and δ such that P p (cid:0) Rad int ( K v ) ≥ r and Rad ext ( K v ) ≤ c r (cid:1) ≤ C e − c r (4.15)for every 0 ≤ p ≤ p c + δ and r ≥
1. It follows in particular that there exist positive constants c r such that P p (cid:0) Rad ext ( K v ) ≥ c r (cid:1) ≥ P p (cid:0) Rad int ( K v ) ≥ r (cid:1) − P p (cid:0) Rad int ( K v ) ≥ r and Rad ext ( K v ) ≤ c r (cid:1) ≥ c r exp (cid:0) − C | p − p c | r (cid:1) − C e − c r ≥ c r exp (cid:0) − C | p − p c | r (cid:1) (4.16)for every p ∈ ( p c − δ , p c ] and r ≥ r . This is easily seen to imply (4.14) in the case p ∈ ( p c − δ , p c ].We now treat the supercritical case. Combining the inequality (4.16) with (3.13), an easyargument similar to that of the previous paragraph shows that there exist positive constants c , C , and C such that P p (cid:16) Rad ext ( K v ) ≥ r and | K v | ≤ C | p − p c | − r (cid:17) ≥ c r exp (cid:0) − C | p − p c | r (cid:1) for every p ∈ ( p c − δ , p c ). We apply a similar coupling argument to the end of the proof ofProposition 4.3, with the important difference that we compare ( p c + ε )-percolation to ( p c − ε )-percolation rather than to p c -percolation. Let 0 < ε ≤ δ , let p = p c + ε , and let q = p c − ε . Let ω p and ω q be Bernoulli- p and Bernoulli- q percolation on G coupled in the standard monotone way,so that, conditional on ω q , every ω q -open edge is ω p open and every ω q -closed edge is chosen tobe either ω p -open or ω p -closed independently at random with probability ( p − q ) / (1 − q ) = O ( ε )to be ω p -open. Let K qv and K pv denote the clusters of v in ω q and ω p respectively. Let A r be theevent that K qv has extrinsic radius at least r and volume at most C ε − r , and let B r be the eventthat K pv is finite and has extrinsic radius at least r . If A r occurs but B r does not, then theremust exist an ω q -closed edge in the boundary of K qv that is ω p -open and whose other endpoint isconnected to infinity in ω p by an open path that does not visit any vertex of K qv . Conditional on K qv , the probability that any particular edge in the boundary of K qv has this property is boundedby ( p − q ) θ ∗ ( p ) / (1 − q ) = O ( ε ), and it follows by the FKG inequality that there exists a constant C such that P ( B r | K qv ) ≥ ( A r ) (cid:20) − ∧ ( p − q ) θ ∗ ( p )1 − q (cid:21) M | K qv | ≥ ( A r ) e − C εr , where M is the maximum degree of G and where we used that | K qv | ≤ C ε − r on the event A r inthe second inequality. Taking expectations, it follows that P ( B r ) ≥ P ( A r ) e − C εr ≥ c r e − ( C + C ) εr and hence that there exists a constant C such that P p (cid:0) r ≤ Rad int ( K v ) < ∞ (cid:1) ≥ P p (cid:0) r ≤ Rad ext ( K v ) < ∞ (cid:1) ≥ c r exp (cid:0) − C | p − p c | r (cid:1) (4.17)for every p ∈ ( p c , p c + δ ) and r ≥
1. This completes the proof. (Note that this argument cannotbe applied directly to the intrinsic radius as written due to non-monotonicity issues.)We now have all the ingredients required to conclude the proofs of our main theorems.
Proof of Theorem 1.1.
The upper bound follows from Propositions 4.1 and 3.7, while the lower37ound follows from Proposition 4.3.
Proof of Theorem 1.2.
The upper bound follows from Propositions 4.1 and 3.4, while the lowerbound follows from Proposition 4.5.
In this subsection we discuss the (apparently rather substantial) challenges that remain to extendour analysis from nonamenable graphs to the high-dimensional Euclidean setting, and give someperspectives on how these challenges might be overcome.Let us begin by stating what is conjectured to be the case. Let d ≥ Z d . The conjectured analogue of Theorem 1.1 is that there exists δ > P p ( n ≤ | K | < ∞ ) ≍ n − / exp h − Θ (cid:0) | p − p c | n (cid:1)i p ∈ ( p c − δ, p c ) n − / p = p c n − / exp (cid:20) − Θ (cid:16)(cid:0) | p − p c | n (cid:1) ( d − /d (cid:17)(cid:21) p ∈ ( p c , p c + δ ) , (5.1)while the conjectured analogue of Theorem 1.2 is that P p ( r ≤ Rad int ( K ) < ∞ ) ≍ n − exp h − Θ (cid:0) | p − p c | n (cid:1)i (5.2)and P p ( r ≤ Rad ext ( K ) < ∞ ) ≍ n − exp (cid:20) − Θ (cid:16) | p − p c | / n (cid:17)(cid:21) (5.3)for all p ∈ ( p c − δ, p c + δ ). Further related questions of interest include the behaviour of the truncatedtwo-point function ˆ τ p ( x, y ) = P ( x ↔ y, x = ∞ ), which is conjectured to satisfyˆ τ p ( x, y ) ≍ k x − y k − d +2 exp (cid:20) − Θ (cid:16) | p − p c | / k x − y k (cid:17)(cid:21) (5.4)for all p ∈ ( p c − δ, p c + δ ) and x, y ∈ Z d . In particular, it is conjectured that the correlationlength ξ ( p ) satisfies ξ ( p ) − := − lim n →∞ n log sup n P p (0 ↔ x, = ∞ ) : x ∈ Z d , k x k ≥ n o ≍ | p − p c | − / (5.5)for p ∈ ( p c − δ, p c + δ ). (Note that (5.4) would trivially imply (5.5).) Besides their intrinsic interest, asolution to these conjectures may be a necessary prerequisite to understanding invasion percolation,the minimal spanning forest, and random walks on slightly supercritical clusters. See [30, Part IV]for an overview.At present, the state of these conjectures can be summarised as follows: The p = p c cases of(5.1) and (5.2) were proven to hold for all quasi-transitive graphs satisfying the triangle conditionby Barsky and Aizenman [6] and Kozma and Nachmias [40], respectively. We showed how thesestatements imply the subcritical cases of the same statements in Propositions 4.1–4.3. Hara and38lade [27] proved via the lace expansion that the triangle condition holds on Z d for sufficientlylarge d , as well as for “spread out” models in dimension d ≥
7. Around the same time, Hara builtupon the methods of [27] to prove the p ≤ p c case of (5.5) under the same hypotheses, i.e., that ξ ( p ) ≍ ( p − p c ) − / as p ↑ p c . Later, Hara, van der Hofstad, and Slade [28] performed a ‘physicalspace’ version of the lace-expansion that allowed them to prove the p = p c case of (5.4) under thesame hypotheses. Kozma and Nachmias [41] then applied this result to prove the p = p c case of(5.3). (It appears that the subcritical cases of (5.3) and (5.4) remain open; we expect that thesecan be handled with existing techniques.) In contrast, almost no progress has been made on theslightly supercritical cases of these conjectures.As we stated in the introduction, we are optimistic that some of the techniques we have de-veloped in this paper will be prove useful to the eventual solution of these conjectures. We nowoutline some ideas about what such a solution might look like. Note that several of the challengesone would need to overcome to adapt our methods to the high-dimensional Euclidean setting areof a similar nature to those one would need to overcome to solve the more qualitative problemsstated in [29, Section 5.3].1. A good first step would be to find a sharp bound on the negative part of the derivative D p,n | K | k for p slightly supercritical. Such a bound would need to be of order C k ( k !) d/ ( d − | p − p c | − k , but it is unclear what form it should take, presumably being written in terms of somehigher truncated moment. A potentially serious difficulty is that it seems one cannot relyon a worst case analysis of the expected number of edges connecting some deterministicset S to infinity off of S , as we did in the proof of Proposition 3.1. Indeed, heuristically,if Λ n = [ − n, n ] d is a box with n = Ω( ξ ( p )) = Ω(( p − p c ) − / ) then the typical numberof edges in the boundary of Λ n whose other endpoint is connected to ∞ off of Λ n shouldbe of order ( p − p c ) / | Λ n | ( d − /d , where ( p − p c ) / is conjectured to be the order of theprobability that the origin is connected to infinity inside a half-space. See [14] for variousrelated rigorous results. This (presumably) worst case bound would be too small to lead toa proof of (5.1), even if one did not have the positive term to contend with. Thus, to bound D p,n via this approach, one would need to somehow understand how the geometry of largefinite clusters in slightly supercritical percolation leads them to have a greater number ofpivotal connections to infinity in their boundary than a box of comparable volume would.The techniques developed to understand phenomena such as Wulff crystals in supercriticalpercolation may be relevant [13].An alternative approach may be to use the OSSS inequality , due to ODonnel, Saks, Schramm,and Servedio [51], which has recently been recognised as a powerful tool in the study ofpercolation and other models following the breakthrough work of Duminil-Copin, Raoufi, andTassion [18,19]; see also [35] for applications to the critical behaviour of Bernoulli percolation.Briefly, this inequality lets us prove differential inequalities by finding randomized algorithmsthat determine the value of the function whose expectation we are interested in but whichhave a low maximum revealment , that is, a low maximum probability of querying whetherany particular edge is open or closed. While this inequality is most powerful as a tool forstudying monotone functions, it can also be used to bound the expected total number ofpivotals for non-monotone functions, which would mean bounding the sum D p,n + U p,n inour context. Such a bound would in fact be just as viable in the remainder of our strategy39s a bound on D p,n itself. The difficulty with this approach is to find, say, a low-revealmentalgorithm determining whether or not the origin is in a large finite cluster. It is unclear howthis might be done. One possibility is to use invasion percolation, but this may be puttingthe cart before the horse; it seems that invasion percolation should be even harder to analysethan slightly supercritical percolation itself.2. Even if one is able to get good bounds on D p,n or D p,n + U p,n , there remains the substantialchallenge of getting good upper bounds on U p,n in the manner of (1.10). It is possible that thiscould be done by methods that are rather similar to what we have done in Sections 3.3–3.5.However, it is likely that, due to the different form of the lower bound on the negative term,one would need to initiate this analysis by proving a version of our skinny clusters estimatein which one could profitably take the radius to be at least a power of the radius rather thana small multiple as we have done here. Bounds of this form are known for Galton-Watsontrees [1, 2], but it seems unclear what one could hope to be true for high dimensional lattices,or how such an estimate might be proven. If such a bound on skinny clusters were found, weare hopeful that an analysis very similar to that performed in Sections 3.4 and 3.5 could beused derive the higher-order variants of this bound needed to bound U p,n | K | k .Finally, we remark that, by analogy with our setting, it may be substantially easier to obtain thecorrect behaviour for the intrinsic radius than for the volume. Acknowledgments
We thank Jonathan Hermon and Asaf Nachmias for many helpful discussions, and thank Remcovan der Hofstad for helpful comments on an earlier version of this manuscript. We also thankAntoine Godin for sharing his simplified proof of Proposition 3.1 with us.
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