Effective resistances for supercritical percolation clusters in boxes
aa r X i v : . [ m a t h . P R ] J un Effective resistances for supercritical percolationclusters in boxes
Yoshihiro Abe ∗ Abstract
Let C n be the largest open cluster for supercritical Bernoulli bondpercolation in [ − n, n ] d ∩ Z d with d ≥
2. We obtain a sharp estimatefor the effective resistance on C n . As an application we show that thecover time for the simple random walk on C n is comparable to n d (log n ) .Noting that the cover time for the simple random walk on [ − n, n ] d ∩ Z d is of order n d log n for d ≥ n (log n ) for d = 2), this givesa quantitative difference between the two random walks for d ≥ MSC 2010 subject classifications : Primary 60J45; Secondary 60K37
Keywords : effective resistances; simple random walks; cover times; Gaussianfree fields; supercritical percolation
The effective resistance is a fundamental measurement of the conductivity forthe electrical network. It has close connections with many subjects of reversibleMarkov chains such as transience/recurrence, heat kernels, mixing times andcover times. We refer to [17], [25], [26], [29] for the introduction of the theoryof reversible Markov chains.Effective resistances for percolation clusters in Z d have been studied for along time. Grimmett, Kesten and Zhang [19] showed almost-sure finiteness ofthe effective resistance from a fixed point to infinity on the infinite supercriticalpercolation cluster in Z d for d ≥
3. (This is equivalent to almost-sure transienceof the simple random walk on the cluster.) The result was extended to a series ofworks on more general energy of flows on percolation clusters [2, 9, 20, 21, 22, 28].The study of effective resistances for boxes on Z d under percolation goes backto 1980’s [11, 18, 23]. These results showed that critical phenomena occur atthe critical percolation probability for effective resistances between opposingfaces of boxes (see Remark 1.3 below). In [10], Boivin and Rau estimated theeffective resistances from a fixed point to boundaries of boxes in the infinite ∗ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan;[email protected] Z (see Remark 1.4 below). In [7], generalupper bounds for effective resistances on general graphs are given by usingisoperimetric inequalities. The estimates are used to obtain upper bounds ofeffective point-to-point resistances on supercritical percolation clusters in boxeson Z ; however, there seems to be a gap in this part (see Remark 1.2 below).Many properties of simple random walks on supercritical percolation clus-ters in Z d have been investigated such as transience [2, 9, 19, 20], mixing times[8, 13], heat kernel decays [4, 31], invariance principles [6, 30, 35], collisions oftwo independent simple random walks [5, 12], and the existence of the harmonicmeasure [10]. These properties are very similar to those of simple random walkson Z d . A famous folk conjecture is that most important properties of simple ran-dom walks survive for simple random walks on supercritical percolation clusters(see, for example, [33]).In this paper, we consider the largest supercritical percolation cluster in[ − n, n ] d ∩ Z d . We obtain the correct order of the maximum of the effectiveresistances between vertices in the giant cluster. Applying the result, we obtaina sharp estimate of the cover time for the simple random walk on the largestcluster; the result shows that it is much larger than the cover time for the simplerandom walk on [ − n, n ] d ∩ Z d when d ≥
3. This gives a negative answer to thefolk conjecture.In order to describe our results more precisely, we begin with some notation.Let | · | be the ℓ -distance on Z d . We define the set of edges between allnearest-neighbour pairs on Z d by E ( Z d ) := {{ x, y } : x, y ∈ Z d , | x − y | = 1 } . For p ∈ [0 , , let P p be the product Bernoulli measure on { , } E ( Z d ) with P p ( ω ( e ) = 1) = 1 − P p ( ω ( e ) = 0) = p for each e ∈ E ( Z d ) . We say that an edge e is open for ω ∈ { , } E ( Z d ) if ω ( e ) = 1. For A ⊆ Z d , we define a random set ofedges by O A = O A ( ω ) := {{ x, y } ∈ E ( Z d ) : x, y ∈ A, ω ( { x, y } ) = 1 } . (1.1)An open cluster in A is a connected component of the graph ( A, O A ) . We definethe critical probability p c ( Z d ) byinf { p ∈ [0 ,
1] : P p (the cluster in Z d containing the origin is infinite) > } . (1.2)We focus our attention to clusters in the box B ( n ) := [ − n, n ] d ∩ Z d . (1.3)It is known that for d ≥ p > p c ( Z d ) , we have the unique largest opencluster in B ( n ) whose size is proportional to n d , P p -a.s., for large n ∈ N (see[14, Theorem 1] for d = 2 and [34, Theorem 1.2] for d ≥ C n todenote the largest open cluster in B ( n ) . Let G = ( V ( G ) , E ( G )) be a finite connected graph; the set V ( G ) is thevertex set and E ( G ) is the edge set. We define the Dirichlet energy by E ( f ) := 12 X u,v ∈ V ( G ) { u,v }∈ E ( G ) ( f ( u ) − f ( v )) , f ∈ R V ( G ) . A, B ⊂ V ( G ) with A ∩ B = ∅ , the effective resistance between A and B for G is defined by R G eff ( A, B ) − := inf {E ( f ) : f ∈ R V ( G ) , E ( f ) < ∞ , f | A = 1 , f | B = 0 } . We write R G eff ( x, y ) to denote R G eff ( { x } , { y } ). We now state our result on theeffective resistance for C n . Theorem 1.1
For d ≥ and p ∈ ( p c ( Z d ) , , there exist c , c > such that P p -a.s., for large n ∈ N , c · log n ≤ max x,y ∈C n R C n eff ( x, y ) ≤ c · log n. Remark 1.2
Corollary 3.1 of [7] says that for d = 2 and p > p c ( Z ) , thereexists c > such that lim n →∞ P p (max x,y ∈C n R C n eff ( x, y ) ≤ c · log n ) = 1 . Theproof is based on general upper bounds for effective resistances given in Theorem2.1 of [7] and an isoperimetric profile for C n studied in [8]. However, accordingto the latest arXiv version of their paper [7] and [24], the proof only impliesthe following: there exists c > such that lim n →∞ P p (max x,y ∈C n R C n eff ( x, y ) ≤ c · (log n ) ) = 1 . This is due to bad isoperimetry of some small subsets of C n . Remark 1.3
Fix d ≥ . Let us consider a random graph G n which has thevertex set [0 , n ] d ∩ Z d and the edge set O [0 ,n ] d ∩ Z d . We write R n to denote theeffective resistance R G n eff ([0 , n ] d − × { } , [0 , n ] d − × { n } ) . In [11, 18], it was shown that the following holds: there exist c , c > suchthat if p < p c ( Z d ) , R n = ∞ for large n ∈ N , P p − a.s., and if p c ( Z d ) < p ≤ , c ≤ lim inf n →∞ n d − R n ≤ lim sup n →∞ n d − R n ≤ c , P p − a.s. Remark 1.4
Let C ∞ be the infinite Bernoulli bond percolation cluster in Z .Define spheres in C ∞ by ∂B C ∞ ( x, r ) = { y ∈ C ∞ : d C ∞ ( x, y ) = r } , x ∈ C ∞ , r ∈ N , where d C ∞ is the graph distance of C ∞ . In Proposition 4.3 of the publishedversion of [10], Boivin and Rau proved the following: for p ∈ ( p c ( Z ) , , thereexist c , c > such that P p -a.s., for all x ∈ C ∞ , for large n ∈ N , c log n ≤ R C ∞ eff ( x , ∂B C ∞ ( x , n )) ≤ c log n. G = ( V ( G ) , E ( G )) and x ∈ V ( G ), let τ x ( G ) be thehitting time of x by the simple random walk on G . We define the cover time by t cov ( G ) := max x ∈ V ( G ) E x (cid:16) max y ∈ V ( G ) τ y ( G ) (cid:17) . Applying Theorem 1.1, we can obtain an improvement of Proposition 3.3 of thearXiv version of [1].
Theorem 1.5
For d ≥ and p ∈ ( p c ( Z d ) , , there exist c , c > such that P p -a.s., for large n ∈ N , c · n d (log n ) ≤ t cov ( C n ) ≤ c · n d (log n ) . Remark 1.6
It is known that t cov ( B ( n )) is comparable to n (log n ) when d = 2 and to n d log n when d ≥ (see, for example, Section 11.3.2 of [26]).Therefore, Theorem 1.5 implies that the simple random walk on C n exhibitsanomalous behavior when d ≥ . This is due to irregularity of C n ; the cluster C n contains a lot of one-dimensional objects (see Lemma 2.6 below). Let G = ( V ( G ) , E ( G )) be any finite connected graph. Let us define theGaussian free field on G . This is a centered Gaussian process { η Gx } x ∈ V ( G ) with η Gx = 0 for some fixed vertex x ∈ V ( G ) and the covariances are given by E ( η Gx η Gy ) = 12 (cid:16) R G eff ( x, x ) + R G eff ( y, x ) − R G eff ( x, y ) (cid:17) , x, y ∈ V ( G ) . We define the expected maximum of the Gaussian free field by M G := E (cid:16) max x ∈ V ( G ) η Gx (cid:17) . Note that M G does not depend on the choice of x . For a set S , we will write | S | to denote the cardinality of S . In [16], Ding, Lee and Peres proved the following:there exist universal constants c , c > c · | E ( G ) | · ( M G ) ≤ t cov ( G ) ≤ c · | E ( G ) | · ( M G ) . (1.4)By (1.4) and Theorem 1.5, we obtain the following estimate of M C n immediately. Corollary 1.7
For d ≥ and p ∈ ( p c ( Z d ) , , there exist c , c > such that P p -a.s., for large n ∈ N , c · log n ≤ M C n ≤ c · log n. Remark 1.8
By Remark 1.6 and (1.4), M B ( n ) is comparable to log n when d = 2 and to √ log n when d ≥ . Thus there is a marked quantitative differencebetween M C n and M B ( n ) when d ≥ . n . InSubsection 3.2, we construct the so-called “Kesten grid” (the terminology comesfrom Mathieu and Remy [31]). This is an analogue of the square lattice andconsists of “white” sites on a renormalized lattice; the concept of “white” sitesis from the renormalization argument of Antal and Pisztora [3]. In Subsection3.3, we prove Theorem 1.1. The lower bound is immediately followed by thefact that C n has a one-way open path of length of order log n . In the proof ofthe upper bound, we construct unit flows between vertices of C n with energyof order log n . Thanks to Kesten grids, we can let the flows run almost onsome “sheets” and adapt an argument of estimating upper bounds of effectiveresistances for Z . We use a result on the chemical distance for C n based on[13]; this guarantees that every vertex of C n can be connected with a Kestengrid decorated with small boxes by an open path in C n of length of order log n .Throughout the paper, we will write c, c ′ , c ′′ to denote positive constantsdepending only on the dimension of the lattice and the percolation parameter.Values of c, c ′ and c ′′ will change from line to line. We use c , c , · · · to denoteconstants whose values are fixed within each argument. If we cite elsewhere theconstant c in Lemma 2.4, we write it as c . . , for example. In this Section, we collect some useful results on percolation estimates. We needthe following facts to prove Theorem 1.1 and Theorem 1.5.(1) Size and connectivity pattern of C n A path is a sequence ( x , x , · · · , x ℓ ) satisfying that | x i − − x i | = 1 for all1 ≤ i ≤ ℓ . An open path is a path all of whose edges are open. We write | · | ∞ to denote the ℓ ∞ -distance on Z d . The diameter of an open path ( x , x , · · · , x ℓ )is defined by max ≤ i,j ≤ ℓ | x i − x j | ∞ . For κ >
0, let A nκ be the event that everyopen path in B ( n ) with diameter larger than κ log n is contained in C n . Thefollowing fact is easily followed by [14, Theorem 1 and Theorem 9] for d = 2and [34, Theorem 1.2 and Theorem 3.1] for d ≥ Lemma 2.1
Fix d ≥ and p > p c ( Z d ) . There exist c , c > and κ , n ∈ N such that for all κ ≥ κ and n ≥ n , P p ( A nκ ∩ {|C n | ≥ c n d } ) ≥ − − c κ log n ) . Remark 2.2
By Lemma 2.1 and the Borel-Cantelli lemma, there exist c , c > such that P p -a.s., for large n ∈ N , the event A nc ∩ {|C n | ≥ c n d } holds.
52) Chemical distance of C n Let d C n be the graph distance of the graph ( C n , O C n ) . The following result isimmediately followed by Lemma 3.2 and (4.1) in [13].
Lemma 2.3
Fix d ≥ and p > p c ( Z d ) . There exist c > and κ ∈ N suchthat the following holds for all κ ≥ κ , P p -a.s., for large n ∈ N : for all x, y ∈ C n with | x − y | ≤ κ log n , d C n ( x, y ) ≤ c κ log n. (3) Crossing probabilitiesIn (3), we restrict our attention to Bernoulli site percolation on the squarelattice. Let Q q be the product Bernoulli measure on { , } Z with Q q ( ω ( x ) =1) = 1 − Q q ( ω ( x ) = 0) = q for each x ∈ Z . For ω ∈ { , } Z and x ∈ Z , wewill say that x is occupied if ω ( x ) = 1 . The critical probability q c ( Z ) is definedsimilarly to (1.2). We write x · y to denote the inner product of x and y . Aself-avoiding path is a path all of whose vertices are distinct. Let e , e be thestandard basis for Z . Fix m, n ∈ N . A crossing in ([0 , m ] × [0 , n ]) ∩ Z is aself-avoiding path ( x , x , · · · , x ℓ ) with x · e = 0 and x ℓ · e = m . Lemma 2.4 [23, Theorem 11.1] For d = 2 and q > q c ( Z ) , there exist c , c , c > such that Q q (cid:16) there exist at least c n disjoint crossingsconsisting of occupied sites in ([0 , m ] × [0 , n ]) ∩ Z (cid:17) ≥ − c · m exp( − c n ) . (4) Renormalization argumentWe recall the renormalization argument of [3]. We will write a, b, · · · rather than x, y, · · · to denote vertices of the renormalized lattice. Fix a positive integer K .To a ∈ Z d , we associate boxes B a ( K ) := (2 K + 1) a + [ − K, K ] d ∩ Z d , (2.1) B ′ a ( K ) := (2 K + 1) a + h − K, K i d ∩ Z d . (2.2)We will say that an open cluster crosses a box if the cluster intersects all thefaces of the box. For a ∈ Z d , we define an event R Ka satisfying the following: • There exists a unique open cluster C in B ′ a ( K ) crossing B ′ a ( K ). • The open cluster C crosses all the subboxes of B ′ a ( K ) of side length largerthan K . • Any open paths in B ′ a ( K ) of diameter larger than K are contained in C . Figure 1: An illustration of a 2-special vertex x and the corresponding specialopen path. The dotted line segments at x + ie ( i = 1 ,
2) are closed edges. Thesolid line segment between x and x + 2 e are composed of open edges.We say a ∈ Z d is white for ω ∈ { , } E ( Z d ) if R Ka ( ω ) = 1 . By [14, Theorem9] for d = 2 and [34, Theorem 3.1] for d ≥
3, if p > p c ( Z d ), we have for all a ∈ Z d lim K →∞ P p ( R Ka ) = 1 . By [27, Theorem 0.0 (ii)], the following holds:
Lemma 2.5
For d ≥ and p > p c ( Z d ) , there exists a function q : N → [0 , with lim K →∞ q ( K ) = 1 such thatthe process ( R Ka ) a ∈ Z d stochastically dominatesa Bernoulli site percolation process in Z d with parameter q ( K ) . (5) Special open pathsLet e , · · · , e d be the standard basis for Z d . We define a one-sided boundary of B ( n ) by ∂ B ( n ) := { x ∈ B ( n ) : x · e = n } . We will say that x ∈ ∂ B ( n ) is m -special if it is a base point of a special openpath; that is, the edge { x + ie , x + ( i + 1) e } is open for each 0 ≤ i ≤ m − x + ie does not have any other open edges for each 1 ≤ i ≤ m .See Figure 1. The following is a key to prove the lower bounds of Theorem 1.1and Theorem 1.5. Lemma 2.6
For d ≥ and p ∈ ( p c ( Z d ) , , there exist c , c > such that P p -a.s., for large n ∈ N , |{ x ∈ ∂ B ( n ) ∩ C n : x is ⌊ c log n ⌋ -special }| ≥ n c . Proof.
Fix a large constant
K >
0. We define a subset of ∂ B ( n ) by ∂ K , B ( n ) := n x ∈ ∂ B ( n ) : | x · e i | ≤ K for all 3 ≤ i ≤ d o . S Kn be the “square” defined by n a ∈ Z d : | a · e | ≤ l n K + 1 m , | a · e | ≤ j n − K/ K + 1 k , a · e i = 0 for 3 ≤ i ≤ d o . Note that S Kn is isomorphic to [0 , ⌈ n K +1 ⌉ ] × [0 , ⌊ n − K/ K +1 ⌋ ] ∩ Z . Let ε > c, c ′ > P p -probability at least 1 − c · n exp( − c ′ n ), we have morethan εn disjoint crossings of white sites in S Kn . By this fact and the definition ofwhite sites together with Lemma 2.1, the event G Kn := {| ∂ K , B ( n ) ∩ C n | < εn } satisfies the following: there exist c, c ′ > κ ∈ N such that for all κ ≥ κ and sufficiently large n ∈ N , P p ( G Kn ) ≤ c exp( − c ′ κ log n ) . (2.3)Take c > r := 1 − c (cid:0) log p + 2( d −
1) log − p (cid:1) positive.Fix 0 < c < r . We write Bin( m, q ) to denote a binomial random variable withparameter m and q . Let q n be the probability of the event that a fixed vertex in ∂ B ( n ) is ⌊ c log n ⌋ -special. Consider the σ -field generated by finite-dimensionalcylinders associated with configurations restricted to B ( n ). Note that condi-tioned on the σ -field, events { x is ⌊ c log n ⌋ -special } and { y is ⌊ c log n ⌋ -special } are independent for x, y ∈ ∂ B ( n ) with | x − y | ∞ ≥
2. By conditioning on the σ -field, we have for some c, c ′ > < ε ′ < ε , P p n |{ x ∈ ∂ B ( n ) ∩ C n : x is ⌊ c log n ⌋ -special }| ≤ n c o ∩ ( G Kn ) c ! ≤ P p Bin (cid:16) ⌊ ε ′ n ⌋ , q n (cid:17) ≤ n c ! ≤ c exp( − c ′ n r ) . (2.4)In the last inequality, we have used the Chebyshev inequality and the fact that nq n ≥ cn r for some c >
0. By (2.3) and (2.4) together with the Borel-Cantellilemma, we obtain the conclusion. (cid:3)
In this Subsection we prove Theorem 1.5 via Theorem 1.1. We begin withgeneral bounds on cover times based on [16, 32]. The following is immediatelyfollowed by [26, Proposition 10.6 and Theorem 11.2] (see also [15, 32]) and [16,Theorem 1.1 and Lemma 1.11]. 8 emma 3.1
Let G = ( V ( G ) , E ( G )) be any finite connected graph.(1) There exists a universal constant c > such that t cov ( G ) ≤ c · | E ( G ) | · (cid:16) max x,y ∈ V ( G ) R G eff ( x, y ) (cid:17) · log | V ( G ) | . (2) There exists a universal constant c > such that for all subset ˜ V ⊂ V ( G ) , t cov ( G ) ≥ c · | E ( G ) | · (cid:16) min x,y ∈ ˜ Vx = y R G eff ( x, y ) (cid:17) · log | ˜ V | . Proof of Theorem 1.5 via Theorem 1.1.
The upper bound is immediately fol-lowed by Lemma 3.1 (1) and Theorem 1.1. From now, we prove the lower bound.By Remark 2.2, we have P p -a.s., for large n ∈ N , C n −⌈ c . . log n ⌉ ⊆ C n . (3.1)Set m n := n − ⌈ c . . log n ⌉ . We define a set V n of tips of special open paths by { x + ⌊ c . . log m n ⌋ e : x ∈ ∂ B ( m n ) ∩ C m n and x is ⌊ c . . log m n ⌋ -special } . By (3.1), Lemma 2.6 and the Nash-Williams inequality (see, for example, [26,Proposition 9.15]), we have the following P p -a.s., for large n ∈ N : for all x, y ∈ V n with x = y , V n ⊆ C n , | V n | ≥ ( m n ) c . . and R C n eff ( x, y ) ≥ ⌊ c . . log m n ⌋ . Therefore, by Remark 2.2 and Lemma 3.1 (2) with ˜ V = V n , we obtain the lowerbound. (cid:3) In this Subsection, we construct the so-called “Kesten grids” on the renormalizedlattice (recall Section 2 (4)) in order to obtain the upper bound of Theorem1.1. The terminology is due to Mathieu and Remy [31]. We note that theconstruction of the Kesten grid is based on Theorem 11.1 of [23] (see Lemma2.4).Let us define some notations. Recall that e , · · · , e d is the standard basis for Z d . Fix a sufficiently large positive integer K . Recall the notation (2.2). For asubset S of the renormalized lattice, we define a fattened version of S by W ( S ) := [ a ∈ S B ′ a ( K ) . (3.2)Let α be a positive constant. We will choose α sufficiently large later. Let ℓ n be the largest integer ℓ satisfying that(2 K + 1)((2 ⌈ α log n ⌉ + 1) ℓ + ⌈ α log n ⌉ ) + 54 K ≤ n, ℓ n := (2 ⌈ α log n ⌉ + 1) ℓ n + ⌈ α log n ⌉ . (3.3)Note that we have W ( B (˜ ℓ n )) ⊂ B ( n ) (recall the notation (1.3)). Thus we canregard B (˜ ℓ n ) as a box in the renormalized lattice corresponding to the originalbox B ( n ). The number of two-dimensional sections in B (˜ ℓ n ) is d ( d − (2˜ ℓ n +1) d − ; each of them is isomorphic to the square [ − ˜ ℓ n , ˜ ℓ n ] ∩ Z . In Subsection3.3 below, we will focus our attention on one of them, namely F := { k e + k e : − ˜ ℓ n ≤ k , k ≤ ˜ ℓ n } . (3.4)We write F i , ≤ i ≤ d ( d − ℓ n + 1) d − (3.5)to denote the other two-dimensional sections of B (˜ ℓ n ). For − ℓ n ≤ m ≤ ℓ n , wedefine the m -th horizontal strip of F R n ( m ) := { k e + k e : − ˜ ℓ n ≤ k ≤ ˜ ℓ n , | k − (2 ⌈ α log n ⌉ + 1) m | ≤ ⌈ α log n ⌉} (3.6)and the m -th vertical strip of F R n ( m ) := { k e + k e : − ˜ ℓ n ≤ k ≤ ˜ ℓ n , | k − (2 ⌈ α log n ⌉ + 1) m | ≤ ⌈ α log n ⌉} . (3.7)We define a horizontal (respectively, vertical) crossing of F as a self-avoidingpath with endvertices a, b satisfying a · e = − ˜ ℓ n and b · e = ˜ ℓ n (respectively, a · e = − ˜ ℓ n and b · e = ˜ ℓ n ). We define strips and crossings for the othertwo-dimensional sections of B (˜ ℓ n ) in a similar fashion. We note that each stripis isomorphic to ([0 , ℓ n ] × [0 , ⌈ α log n ⌉ ]) ∩ Z . Taking α large enough, thefollowing holds immediately by Lemma 2.4 and Lemma 2.5. Corollary 3.2
Fix d ≥ and p > p c ( Z d ) . The following holds P p -a.s., for large n ∈ N : for all ≤ i ≤ d ( d − (2˜ ℓ n + 1) d − and − ℓ n ≤ m ≤ ℓ n , there exist at least c . . ⌈ α log n ⌉ disjoint horizontal (respectively, vertical) crossings consisting ofwhite sites in the m -th horizontal (respectively, vertical) strip of F i . Set L n := (cid:6) c . . ⌈ α log n ⌉ (cid:7) . Fix 1 ≤ i ≤ d ( d − (2˜ ℓ n + 1) d − and a configurationsatisfying the event of Corollary 3.2. We can choose L n disjoint crossings ofwhite sites in each strip of F i . Since F i is isomorphic to [ − ˜ ℓ n , ˜ ℓ n ] ∩ Z , thehorizontal and the vertical crossings intersect (see Figure 2) and form a grid on F i . We will call it a Kesten grid. Note that Kesten grids on F i and F j do notintersect if i = j in general. In this Subsection we prove Theorem 1.1. To prove it, we only need to estimatethe upper bound of effective resistances for pairs of vertices on W ( F i ) ∩ C n , ≤ i ≤ d ( d − (2˜ ℓ n + 1) d − (recall the notations (3.2)-(3.5)).10 d(cid:11) log ne2d(cid:11) log ne2~`n2~`n Figure 2: An illustration of disjoint crossings of white sites (thin solid lines) instrips (rectangles with dotted boundaries) of F (square with thick solid bound-ary). These horizontal and vertical crossings intersect since F is isomorphic to[ − ˜ ℓ n , ˜ ℓ n ] ∩ Z . Proposition 3.3
Fix d ≥ and p > p c ( Z d ) . There exists c > such that P p -a.s., for large n ∈ N , the following holds: for all ≤ i ≤ d ( d − (2˜ ℓ n + 1) d − and x, y ∈ W ( F i ) ∩ C n , R C n eff ( x, y ) ≤ c · log n. Proof of Theorem 1.1 via Proposition 3.3.
Since the triangle inequality holds forthe effective resistance (see, for instance, [26, Corollary 10.8]), the upper boundis immediately followed by using Lemma 2.3 and Proposition 3.3 repeatedly.The lower bound holds by Lemma 2.6, (3.1) and the Nash-Williams inequality. (cid:3)
In the proof of Proposition 3.3 below, we will focus our attention to W ( F ) ∩C n .Recall the constant α in Subsection 3.2. To k , k ∈ Z , we associate the “square”defined by S nk ,k := { s e + s e : | s i − (2 ⌈ α log n ⌉ + 1) k i | ≤ ⌈ α log n ⌉ , i = 1 , } . Fix some integers r, r ′ with r < r ′ . A left-to-right (respectively, right-to-left)path in [ r, r ′ ] × Z d − is a path in [ r, r ′ ] × Z d − whose initial and final vertices a, b satisfy a · e = r and b · e = r ′ (respectively, a · e = r ′ and b · e = r ). Abottom-to-top path in Z × [ r, r ′ ] × Z d − is a path in Z × [ r, r ′ ] × Z d − whoseinitial and final vertices a, b satisfy a · e = r and b · e = r ′ . For a path Γ and a, a ′ ∈ Γ (we assume that a appears before a ′ in Γ), we write Γ[ a, a ′ ] to denotea part of Γ from a to a ′ . In particular, when a ′ is the final vertex of Γ, we write11 m1; ~m2)(m1; m2) (cid:0)i hx hy (cid:0)(i; j) y ~yx~x (cid:13)(i; j)(m1 + i; m2+ ~m22 )Snm1;m2Snm1; ~m2 W (Snm1; ~m2) W (Snm1;m2) Figure 3: Illustrations of paths Γ i , Γ( i, j ) and γ ( i, j ).Γ[ a ] in the place of Γ[ a, a ′ ]. Proof of Proposition 3.3.
We fix a configuration satisfying the events of Re-mark 2.2, Corollary 3.2, and Lemma 2.3 with κ large enough. Let m , m , ˜ m beintegers satisfying the following: m < ˜ m , ˜ m − m is even, and m + ˜ m − m ≤ ℓ n (recall the definition of ℓ n below (3.2)). For simplicity, we only estimate theeffective resistance between x ∈ W ( S nm ,m ) ∩ C n and y ∈ W ( S nm , ˜ m ) ∩ C n . Formore general cases, we can apply a similar argument and use Lemma 2.3. Weomit the details. We will construct a unit flow between x and y . Our argumentis based on [29, Proposition 2.15] and Section 4.1 of the arXiv version of [10].We first construct a random self-avoiding open path from x to y , most ofwhose parts lie on W ( F ) ∩ C n . Recall the notations (3.6) and (3.7). A Kestengrid guarantees the existence of the following self-avoiding paths: • disjoint left-to-right (respectively, right-to-left) self-avoiding paths of whitesites H m , · · · , H mL n contained in R n ( m + m ) for 1 ≤ m ≤ ˜ m − m (respec-tively, ˜ m − m < m < ˜ m − m ), • disjoint bottom-to-top self-avoiding paths of white sites V m , · · · , V mL n con-tained in R n ( m + m ) for 0 ≤ m ≤ ˜ m − m , • left-to-right self-avoiding paths of white sites H x and H y of length at most c log n for some c >
0, contained in S nm ,m and in S nm , ˜ m respectively.From now, we construct an open path from x to y corresponding to a fixed label( i, j ) with i ∈ { , · · · , ˜ m − m } and j ∈ { , · · · , L n } by the following three steps.Step 1: For convenience, we will associate a vertex ( k , k ) ∈ Z to S nk ,k .Let f : { , · · · , i } → { , · · · , i } be a function defined by f ( r ) = r for 0 ≤ r ≤ i and f ( r ) = 2 i − r for i < r ≤ i . Take a self-avoiding path Γ i as follows:12 srj [a2r℄ V f(r)ja2r+1 Hsr+1j V f(r)j [a2r+1℄a2r+2 Figure 4: Illustrations of a r +1 and a r +2 for 0 ≤ r ≤ i . • Let ( s r ) ≤ r ≤ i +1 be a sequence with 0 = s < s < · · · < s i +1 = ˜ m − m .For 0 ≤ r ≤ i , set v r = ( m + f ( r ) , m + s r ) and v r +1 = ( m + f ( r ) , m + s r +1 ). The self-avoiding path Γ i is obtained from the sequence( v r ) ≤ r ≤ i +1 by linear interpolation. • The path Γ i lies within 1 in ℓ ∞ -distance from the piecewise line segmentsconnecting ( m , m ) , ( m + i, m + ˜ m ) and ( m , ˜ m ).See the left-hand side of Figure 3.Step 2: Recall the notation before the proof of Proposition 3.3. We definevertices a , · · · , a i +2 as follows: • For 0 ≤ r ≤ i , a r +1 is the last-visited vertex of H s r j [ a r ] by V f ( r ) j ,where a := h x and H s j := H x . • For 0 ≤ r ≤ i , a r +2 is the last-visited vertex of V f ( r ) j [ a r +1 ] by H s r +1 j ,where H s i +1 j := H y .See Figure 4. We can obtain a self-avoiding path Γ( i, j ) of white sites from h x to h y in ∪ ( k ,k ) ∈ Γ i S nk ,k by connecting segments H s r j [ a r , a r +1 ] , V f ( r ) j [ a r +1 , a r +2 ] , ≤ r ≤ i, and H y [ a i +2 , h y ] . See the middle of Figure 3.Step 3: Recall the notation (2.1). Fix vertices ˜ x ∈ B h x ( K ) ∩ C n and ˜ y ∈ B h y ( K ) ∩ C n . Indeed, we can take such ˜ x and ˜ y by Remark 2.2. By the defi-nition of white sites, we can take a self-avoiding open path ˜ γ ( i, j ) from ˜ x to ˜ y contained in W (Γ( i, j )). By Lemma 2.3, we have self-avoiding open paths γ x and γ y in C n from x to ˜ x and from ˜ y to y respectively of length at most c log n for some c >
0. We can get a self-avoiding open path γ ( i, j ) from x to y in C n via γ x , ˜ γ ( i, j ) and γ y . See the right-hand side of Figure 3.13et X and Y be random variables distributed uniformly on { , · · · , ˜ m − m } and on { , · · · , L n } respectively; they are defined on a probability space withprobability measure P . Recall the notation (1.1). From now, we construct aunit flow from x to y via the random open path γ ( X, Y ). We define a randomfunction ψ on C n × C n by ψ ( u, v ) := { u, v } ∈ O C n and γ ( X, Y ) passes u, v in this order , − { u, v } ∈ O C n and γ ( X, Y ) passes v, u in this order , . We define a function θ on C n × C n by θ ( u, v ) := E ( ψ ( u, v )) for u, v ∈ C n . Since ψ is a unit flow from x to y , P -a.s., θ is a unit flow from x to y . In orderto bound θ , let us define the following function on C n × C n : p ( u, v ) := ( P (cid:0) γ ( X, Y ) passes the edge { u, v } (cid:1) if { u, v } ∈ O C n , . For 0 ≤ r ≤ ˜ m − m , we define a set of labels of the r -th level by D r := ( { ( m + s, m + r ) ∈ Z : 0 ≤ s ≤ r + 1 } if r ≤ ˜ m − m , { ( m + s, m + r ) ∈ Z : 0 ≤ s ≤ ˜ m − m − r + 1 } otherwise . Let U be the union of γ x , γ y , W ( H x ) and W ( H y ). By Thomson’s principle (see,for example, [26, Theorem 9.10]) and the construction of γ ( X, Y ), we have R C n eff ( x, y ) ≤ X u,v ∈C n θ ( u, v ) ≤ X u,v ∈ U p ( u, v ) + ( ˜ m − m ) / X r =0 X ( k ,k ) ∈ D r X u,v ∈ W ( S nk ,k ) v / ∈ U p ( u, v ) + ˜ m − m X r =( ˜ m − m ) / X ( k ,k ) ∈ D r X u,v ∈ W ( S nk ,k ) v / ∈ U p ( u, v ) . (3.8)Since γ x , γ y , H x and H y are self-avoiding paths of length of order log n , thefirst term on the right-hand side of (3.8) is bounded by c log n for some c > ≤ r ≤ ˜ m − m , ( k , k ) ∈ D r and u, v ∈ W ( S nk ,k ) with v / ∈ U and { u, v } ∈ O C n . By the construction of the random open path γ ( X, Y ), we havefor some c > p ( u, v ) ≤ cr · L n .
14o we have for some c, c ′ > ( ˜ m − m ) / X r =0 X ( k ,k ) ∈ D r X u,v ∈ W ( S nk ,k ) v / ∈ U p ( u, v ) ≤ c ( ˜ m − m ) / X r =1 r ≤ c ′ log n. By a similar argument, the last term on the right-hand side of (3.8) is boundedby c log n for some c >
0. Therefore by (3.8), we obtain the conclusion. (cid:3)
Acknowledgements.
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