Effective resistances for supercritical percolation clusters in boxes

www.imstat.org/aihp 2015, Vol. 51, No. 3, 935–946

DOI:10.1214/14-AIHP604

© Association des Publications de l’Institut Henri Poincaré, 2015

Effective resistances for supercritical percolation clusters in boxes ¹

Yoshihiro Abe

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail:[email protected] Received 24 June 2013; revised 12 December 2013; accepted 22 January 2014

Abstract. LetCⁿbe the largest open cluster for supercritical Bernoulli bond percolation in[−n, n]^d∩Z^dwithd≥2. We obtain a sharp estimate for the effective resistance onCⁿ. As an application we show that the cover time for the simple random walk on Cⁿis comparable ton^d(logn)². Noting that the cover time for the simple random walk on[−n, n]^d∩Z^dis of ordern^dlognfor d≥3 (and of ordern²(logn)²ford=2), this gives a quantitative difference between the two random walks ford≥3.

Résumé. On considère la percolation de Bernoulli par arêtes dans le régime surcritique. SoitCⁿle plus grand amas de percolation dans[−n, n]^d∩Z^davecd≥2. Nous obtenons une estimation précise de la résistance effective surCⁿ. Comme application, nous montrons que le temps de recouvrement d’une marche simple surCⁿest de l’ordre den^d(logn)². En remarquant que le temps de recouvrement d’une marche simple sur[−n, n]^d∩Z^dest de l’ordre den^dlognquandd≥3 (et den²(logn)²quandd=2), ceci montre une différence quantitative entre les deux marches sid≥3.

MSC:Primary 60J45; secondary 60K37

Keywords:Effective resistances; Simple random walks; Cover times; Gaussian free fields; Supercritical percolation

1. Introduction

The effective resistance is a fundamental measurement of the conductivity for the electrical network. It has close connections with many subjects of reversible Markov chains such as transience/recurrence, heat kernels, mixing times and cover times. We refer to [17,25,26,29] for the introduction of the theory of reversible Markov chains.

Effective resistances for percolation clusters inZ^dhave been studied for a long time. Grimmett, Kesten and Zhang [19] showed almost-sure finiteness of the effective resistance from a fixed point to infinity on the infinite supercritical percolation cluster inZ^d ford ≥3. (This is equivalent to almost-sure transience of the simple random walk on the cluster.) The result was extended to a series of works on more general energy of flows on percolation clusters [2,9,20–

22,28]. The study of effective resistances for boxes onZ^d under percolation goes back to 1980’s [11,18,23]. These results showed that critical phenomena occur at the critical percolation probability for effective resistances between opposing faces of boxes (see Remark1.3below). In [10], Boivin and Rau estimated the effective resistances from a fixed point to boundaries of boxes in the infinite supercritical percolation cluster onZ²(see Remark1.4below). In [7], general upper bounds for effective resistances on general graphs are given by using isoperimetric inequalities. The estimates are used to obtain upper bounds of effective point-to-point resistances on supercritical percolation clusters in boxes onZ²; however, there seems to be a gap in this part (see Remark1.2below).

1Supported by JSPS Research Fellowships.

Many properties of simple random walks on supercritical percolation clusters inZ^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 of two independent simple random walks [5,12], and the existence of the harmonic measure [10]. These properties are very similar to those of simple random walks onZ^d. A famous folk conjecture is that most important properties of simple random 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 effective resistances between vertices in the giant cluster. Applying the result, we obtain a sharp estimate of the cover time for the simple random walk on the largest cluster; the result shows that it is much larger than the cover time for the simple random walk on[−n, n]^d∩Z^dwhend≥3. The author should mention that the referee pointed out that this result was a conjecture.

In order to describe our results more precisely, we begin with some notation. Let| · |1be the¹-distance onZ^d. We define the set of edges between all nearest-neighbour pairs onZ^dbyE(Z^d):= {{x, y}: x, y∈Z^d,|x−y|1=1}.

Forp∈ [0,1], letPp be the product Bernoulli measure on{0,1}^E(Zd) withPp(ω(e)=1)=1−Pp(ω(e)=0)=p for eache∈E(Z^d). We say that an edgeeis open forω∈ {0,1}^E(Zd) ifω(e)=1. ForA⊆Z^d, we define a random set of edges by

OA=OA(ω):=

{x, y} ∈E Z^d

: x, y∈A, ω {x, y}

=1

. (1.1)

An open cluster inAis a connected component of the graph(A,OA). We define the critical probabilityp_c(Z^d)by inf

p∈ [0,1]: Pp

the cluster inZ^dcontaining the origin is infinite

>0

. (1.2)

We focus our attention to clusters in the box

B(n):= [−n, n]^d∩Z^d. (1.3)

It is known that ford≥2 andp > p_c(Z^d), we have the unique largest open cluster inB(n)whose size is proportional ton^d,Pp-a.s., for largen∈N. Ford≥3, see [34], Theorem 1.2. Thed=2 case is well-known, and [14] will serve as a convenient reference. We writeCⁿto denote the largest open cluster inB(n).

LetG=(V (G), E(G))be a finite connected graph; the setV (G)is the vertex set andE(G)is the edge set. We define the Dirichlet energy by

E(f ):=1 2

u,v∈V (G) {u,v}∈E(G)

f (u)−f (v)2

, f ∈R^{V (G)}.

ForA, B⊂V (G)withA∩B=∅, the effective resistance betweenAandBforGis defined by R_eff^G(A, B)⁻¹:=inf

E(f ): f∈R^{V (G)},E(f ) <∞, f|A=1, f|B=0 .

We writeR_eff^G(x, y)to denoteR^G_eff({x},{y}). We now state our result on the effective resistance forCⁿ. Theorem 1.1. Ford≥2andp∈(p_c(Z^d),1),there existc₁, c₂>0such thatPp-a.s.,for largen∈N,

c₁·logn≤ max

x,y∈CⁿR^C_effⁿ(x, y)≤c₂·logn.

Remark 1.2. Corollary 3.1 of [7] says that for d = 2 and p > p_c(Z²), there exists c₁ > 0 such that limn→∞Pp(maxx,y∈CⁿR_eff^Cn(x, y)≤c₁·logn)=1. The proof is based on general upper bounds for effective re- sistances given in Theorem2.1of[7]and an isoperimetric profile forCⁿ studied in[8].However,according to the latest arXiv version of their paper[7]and [24],the proof only implies the following:there exists c₁>0 such that limn→∞Pp(max_x,y∈CⁿR_eff^Cn(x, y)≤c₁·(logn)²)=1.This is due to bad isoperimetry of some small subsets ofCⁿ.

Remark 1.3. Fixd≥2.Let us consider a random graphG_nwhich has the vertex set[0, n]^d∩Z^d and the edge set O_[0,n]^d∩Z^d.We writeRnto denote the effective resistance

R^G_effⁿ

[0, n]^d−1× {0},[0, n]^d−1× {n} .

In[11,18],it was shown that the following holds:there existc1, c2>0such that ifp < pc(Z^d), R_n= ∞ for largen∈N,Pp-a.s.,

and ifp_c(Z^d) < p≤1, c1≤lim inf

n→∞ n^d−2Rn≤lim sup

n→∞ n^d−2Rn≤c2, Pp-a.s.

Remark 1.4. LetC∞be the infinite Bernoulli bond percolation cluster inZ².Define spheres inC∞by

∂B_C∞(x, r)=

y∈C_∞: d_C∞(x, y)=r

, x∈C_∞, r∈N,

whered_C∞is the graph distance ofC_∞.In Proposition4.3of the published version of[10],Boivin and Rau proved the following:forp∈(p_c(Z²),1],there existc₁, c₂>0such thatPp-a.s.,for allx₀∈C_∞,for largen∈N,

c₁logn≤R^C_eff^∞

x₀, ∂B_C∞(x₀, n)

≤c₂logn.

For a finite connected graphG=(V (G), E(G))andx∈V (G), letτ_x(G)be the hitting time ofx by the simple random walk onG. We define the cover time by

t_cov(G):= max

x∈V (G)E_x

y∈maxV (G)τ_y(G)

.

Applying Theorem1.1, we can obtain an improvement of Proposition 3.3 of the arXiv version of [1].

Theorem 1.5. Ford≥2andp∈(p_c(Z^d),1),there existc₁, c₂>0such thatPp-a.s.,for largen∈N, c1·n^d(logn)²≤tcov

Cⁿ

≤c2·n^d(logn)².

Remark 1.6. It is known thattcov(B(n))is comparable ton²(logn)² whend=2and ton^dlognwhend≥3 (see, for example,Section 11.3.2of [26]).Therefore,Theorem1.5implies that the simple random walk on Cⁿ exhibits anomalous behavior whend≥3.This is due to irregularity ofCⁿ;the clusterCⁿcontains a lot of one-dimensional objects(see Lemma2.6below).

LetG=(V (G), E(G))be any finite connected graph. Let us define the Gaussian free field onG. This is a centered Gaussian process{η^G_x}x∈V (G)withη^G_x

0=0 for some fixed vertexx0∈V (G)and the covariances are given by E

η^G_xη^G_y

=1 2

R^G_eff(x, x₀)+R_eff^G(y, x₀)−R^G_eff(x, y)

, x, y∈V (G).

We define the expected maximum of the Gaussian free field by MG:=E

x∈maxV (G)η^G_x

.

Note thatM_Gdoes not depend on the choice ofx₀. For a setS, we will write|S|to denote the cardinality ofS. In [16], Ding, Lee and Peres proved the following: there exist universal constantsc₁, c₂>0 such that

c₁· E(G) ·(M_G)²≤t_cov(G)≤c₂· E(G) ·(M_G)². (1.4) By (1.4) and Theorem1.5, we obtain the following estimate ofM_Cⁿimmediately.

Corollary 1.7. Ford≥2andp∈(p_c(Z^d),1),there existc₁, c₂>0such thatPp-a.s.,for largen∈N, c₁·logn≤M_Cn≤c₂·logn.

Remark 1.8. By Remark1.6and(1.4),MB(n)is comparable tolognwhend=2and to√

lognwhend≥3.Thus there is a marked quantitative difference betweenM_CⁿandM_B(n)whend≥3.

Let us describe the outline of the paper. Section2gives preliminary lemmas about percolation estimates. In Sec- tion3.1, we give a proof of Theorem1.5via Theorem1.1. The upper bound is due to Theorem1.1and the Matthews bound [32]. The lower bound is based on the general bound on the cover time via the Gaussian free field given in [16].

The key of the proof of the lower bound is counting the number of one-way open paths of length of order logn. In Section3.2, we construct the so-called “Kesten grid” (the terminology comes from Mathieu and Remy [31]). This is an analogue of the square lattice and consists of “white” sites on a renormalized lattice; the concept of “white” sites is from the renormalization argument of Antal and Pisztora [3]. In Section3.3, we prove Theorem1.1. The lower bound is immediately followed by the fact thatCⁿhas a one-way open path of length of order logn. In the proof of the upper bound, we construct unit flows between vertices ofCⁿ with energy of order logn. Thanks to Kesten grids, we can let the flows run almost on some “sheets” and adapt an argument of estimating upper bounds of effective resistances forZ². We use a result on the chemical distance forCⁿ based on [13]; this guarantees that every vertex ofCⁿcan be connected with a Kesten grid decorated with small boxes by an open path inCⁿof length of order logn.

Throughout the paper, we will writec, c, cto denote positive constants depending only on the dimension of the lattice and the percolation parameter. Values ofc, candcwill change from line to line. We usec₁, c₂, . . .to denote constants whose values are fixed within each argument. If we cite elsewhere the constantc₁in Lemma2.4, we write it asc_2.4.1, for example.

2. Percolation estimates

In this section, we collect some useful results on percolation estimates. We need the following facts to prove Theo- rem1.1and Theorem1.5.

(1) Size and connectivity pattern ofCⁿ

A path is a sequence(x₀, x₁, . . . , x)satisfying that|x_i−1−x_i|1=1 for all 1≤i≤. An open path is a path all of whose edges are open. We write| · |_∞to denote the^∞-distance onZ^d. The diameter of an open path(x₀, x₁, . . . , x) is defined by max0≤i,j≤|x_i−x_j|∞. Forκ >0, letAⁿ_κbe the event that every open path inB(n)with diameter larger thanκlognis contained inCⁿ. The following fact follows easily from [34], Theorem 1.2 and Theorem 3.1, ford≥3 and [14], Theorem 1 and Theorem 9, ford=2.

Lemma 2.1. Fixd≥2andp > p_c(Z^d).There existc₁, c₂>0andκ₀, n₀∈Nsuch that for allκ≥κ₀andn≥n₀, Pp

Aⁿ_κ∩ Cⁿ ≥c₁n^d

≥1−2 exp(−c₂κlogn).

Remark 2.2. By Lemma2.1and the Borel–Cantelli lemma,there existc1, c2>0such thatPp-a.s.,for largen∈N, the eventAⁿ_c

1∩ {|Cⁿ| ≥c2n^d}holds.

(2) Chemical distance ofCⁿ

Letd_Cn be the graph distance of the graph(Cⁿ,O_Cⁿ). The following result is immediately followed by Lemma 3.2 and (4.1) in [13].

Lemma 2.3. Fixd≥2andp > p_c(Z^d).There existc1>0andκ0∈Nsuch that the following holds for allκ≥κ0, Pp-a.s.,for largen∈N:for allx, y∈Cⁿwith|x−y|1≤κlogn,

d_Cn(x, y)≤c₁κlogn.

(3) Crossing probabilities

In (3), we restrict our attention to Bernoulli site percolation on the square lattice. Let Qq be the product Bernoulli measure on{0,1}^Z2 withQq(ω(x)=1)=1−Qq(ω(x)=0)=q for eachx∈Z². Forω∈ {0,1}^Z2 andx∈Z², we will say thatx is occupied ifω(x)=1. The critical probabilityq_c(Z²)is defined similarly to (1.2). We writex·y to denote the inner product ofx andy. A self-avoiding path is a path all of whose vertices are distinct. Lete₁, e₂ be the standard basis for Z². Fix m, n∈N. A horizontal crossing in([0, m] × [0, n])∩Z² is a self-avoiding path (x0, x1, . . . , x)withx0·e1=0 andx·e1=m.

Lemma 2.4 ([23], Theorem 11.1). Ford=2andq > q_c(Z²),there existc₁, c₂, c₃>0such that Qq

there exist at leastc₁ndisjoint horizontal crossings

consisting of occupied sites in

[0, m] × [0, n]

∩Z²

≥1−c₂·mexp(−c₃n).

(4) Renormalization argument

We recall the renormalization argument of [3]. We will writea, b, . . .rather than x, y, . . .to denote vertices of the renormalized lattice. Fix a positive integerK. Toa∈Z^d, we associate boxes

B_a(K):=(2K+1)a+ [−K, K]^d∩Z^d, (2.1)

B_a(K):=(2K+1)a+

−5 4K,5

4K d

∩Z^d. (2.2)

We will say that an open cluster crosses a box if the cluster intersects all the faces of the box. Fora∈Z^d, we define an eventR_a^Ksatisfying the following:

• There exists a unique open clusterCinB_a(K)crossingB_a(K).

• The open clusterCcrosses all the subboxes ofB_a(K)of side length larger than₁₀^K.

• Any open paths inB_a(K)of diameter larger than ₁₀^K are contained inC. We saya∈Z^d is white forω∈ {0,1}^E(Zd) if1_RK

a(ω)=1. By [34], Theorem 3.1, ford≥3 and [14], Theorem 9, ford=2, ifp > p_c(Z^d), we have for alla∈Z^d

Klim→∞Pp

R_a^K

=1.

By [27], Theorem 0.0(ii), the following holds:

Lemma 2.5. Ford≥2andp > p_c(Z^d),there exists a functionq:N→ [0,1]withlimK→∞q(K)=1such that the process(1_RK

a )_a∈Zd stochastically dominates

a Bernoulli site percolation process inZ^dwith parameterq(K).

(5) Special open paths

Lete₁, . . . , e_d be the standard basis forZ^d. We define a one-sided boundary ofB(n)by

∂₁B(n):=

x∈B(n): x·e₁=n .

We will say thatx∈∂₁B(n)ism-special if it is a base point of a special open path; that is, the edge{x+ie₁, x+(i+ 1)e1}is open for each 0≤i≤m−1 and the vertexx+ie₁does not have any other open edges for each 1≤i≤m.

See Figure1. The following is a key to prove the lower bounds of Theorem1.1and Theorem1.5.

Fig. 1. An illustration of a 2-special vertexxand the corresponding special open path. The dotted line segments atx+ie₁(i=1,2) are closed edges. The solid line segment betweenxandx+2e₁are composed of open edges.

Lemma 2.6. Ford≥2andp∈(pc(Z^d),1),there existc1, c2>0such thatPp-a.s.,for largen∈N, x∈∂₁B(n)∩Cⁿ: x isc₁logn-special ≥n^c2.

Proof. Fix a large constantK >0. We define a subset of∂₁B(n)by

∂_1,2^KB(n):=

x∈∂₁B(n): |x·e_i| ≤5

4Kfor all 3≤i≤d

.

LetS_n^Kbe the “square” defined by

a∈Z^d: |a·e1| ≤ n

2K+1

,|a·e2| ≤

n−5K/4 2K+1

, a·e_i=0 for 3≤i≤d

.

Note thatS_n^Kis isomorphic to[0,2_2Kⁿ₊₁] × [0,2ⁿ_2K^−5K/4₊₁ ] ∩Z². Letε >0 be a small constant. By Lemma2.4and Lemma2.5, the following holds: there existc, c>0 such that withPp-probability at least 1−c·nexp(−cn), we have more thanεndisjoint crossings of white sites inS_n^K. By this fact and the definition of white sites together with Lemma2.1, the eventG^K_n := {|∂_1,2^K B(n)∩Cⁿ|< εn}satisfies the following: there existc, c>0 andκ0∈Nsuch that for allκ≥κ0and sufficiently largen∈N,

Pp

G^K_n

≤cexp

−cκlogn

. (2.3)

Take c1>0 small enough to make r :=1−c1(log_p¹ +2(d −1)log₁₋¹_p) positive. Fix 0< c2< r. We write Bin(m, q) to denote a binomial random variable with parametermand q. Let qn be the probability of the event that a fixed vertex in∂1B(n)isc1logn-special. Consider theσ-field generated by finite-dimensional cylinders as- sociated with configurations restricted toB(n). Note that conditioned on theσ-field, events{xisc1logn-special}

and{yisc1logn-special}are independent forx, y∈∂1B(n)with|x−y|_∞≥2. By conditioning on theσ-field, we have for somec, c>0 and some 0< ε< ε,

Pp x∈∂₁B(n)∩Cⁿ: xisc₁logn-special ≤n^c2

∩ G^K_nc

≤Pp

Bin εn

, q_n

≤n^c2

≤cexp

−cn^r

. (2.4)

In the last inequality, we have used the Chebyshev inequality and the fact thatnq_n≥cn^r for somec >0. By (2.3) and

(2.4) together with the Borel–Cantelli lemma, we obtain the conclusion.

3. Proof of Theorem1.1and Theorem1.5

3.1. Cover time estimate

In this subsection we prove Theorem1.5via Theorem1.1. We begin with general bounds on cover times based on [16,32]. The following is immediately followed by [26], Proposition 10.6 and Theorem 11.2 (see also [15,32]) and [16], Theorem 1.1 and Lemma 1.11.

Lemma 3.1. LetG=(V (G), E(G))be any finite connected graph.

(1) There exists a universal constantc₁>0such that t_cov(G)≤c₁· E(G) ·

x,ymax∈V (G)R_eff^G(x, y)

·log V (G) .

(2) There exists a universal constantc2>0such that for all subsetV˜⊂V (G), t_cov(G)≥c₂· E(G) ·

min

x,y∈ ˜V x=y

R^G_eff(x, y)

·log| ˜V|.

Proof of Theorem1.5via Theorem1.1. The upper bound is immediately followed by Lemma3.1(1) and Theo- rem1.1. From now, we prove the lower bound. By Remark2.2, we havePp-a.s., for largen∈N,

C^{n−c2.6.1logn}⊆Cⁿ. (3.1)

Setm_n:=n− c_2.6.1logn. We define a setV_nof tips of special open paths by x+ c2.6.1logmne1: x∈∂1B(mn)∩C^mnandxisc2.6.1logmn-special

.

By (3.1), Lemma2.6and the Nash–Williams inequality (see, for example, [26], Proposition 9.15), we have the fol- lowingPp-a.s., for largen∈N: for allx, y∈V_nwithx=y,

Vn⊆Cⁿ, |Vn| ≥(mn)^c2.6.2 and R_eff^Cn(x, y)≥ c2.6.1logmn.

Therefore, by Remark2.2and Lemma3.1(2) withV˜ =V_n, we obtain the lower bound.

3.2. Kesten grid

In this subsection, we construct the so-called “Kesten grids” on the renormalized lattice (recall Section2(4)) in order to obtain the upper bound of Theorem 1.1. The terminology is due to Mathieu and Remy [31]. We note that the construction of the Kesten grid is based on Theorem 11.1 of [23] (see Lemma2.4).

Let us define some notations. Recall thate₁, . . . , e_d is the standard basis forZ^d. Fix a sufficiently large positive integerK. Recall the notation (2.2). For a subsetSof the renormalized lattice, we define a fattened version ofSby

W (S):=

a∈S

B_a(K). (3.2)

Letαbe a positive constant. We will chooseαsufficiently large later. Letnbe the largest integersatisfying that (2K+1)

2αlogn +1

+ αlogn +5

4K≤n, and set

˜n:=

2αlogn +1

n+ αlogn. (3.3)

Note that we haveW (B(˜_n))⊂B(n)(recall the notation (1.3)). Thus we can regardB(˜_n)as a box in the renormalized lattice corresponding to the original boxB(n). The number of two-dimensional sections inB(˜_n)is ^d(d₂⁻¹⁾(2˜_n+ 1)^d−2; each of them is isomorphic to the square[− ˜_n,˜_n]²∩Z². In Section3.3below, we will focus our attention on one of them, namely

F₁:= {k₁e₁+k₂e₂: − ˜_n≤k₁, k₂≤ ˜_n}. (3.4)

We write

F_i, 2≤i≤d(d−1)

2 (2˜_n+1)^d−2 (3.5)

to denote the other two-dimensional sections ofB(˜_n). For−_n≤m≤_n, we define themth horizontal strip ofF₁ R₁ⁿ(m):=

k₁e₁+k₂e₂: − ˜_n≤k₁≤ ˜_n, k₂−

2αlogn +1m ≤ αlogn

(3.6) and themth vertical strip ofF₁

R₂ⁿ(m):=

k₁e₁+k₂e₂: − ˜_n≤k₂≤ ˜_n, k₁−

2αlogn +1

m ≤ αlogn

. (3.7)

We define a horizontal (respectively, vertical) crossing ofF₁as a self-avoiding path with endverticesa, bsatisfying a·e₁= − ˜_n andb·e₁= ˜_n (respectively, a·e₂= − ˜_n andb·e₂= ˜_n). We define strips and crossings for the other two-dimensional sections ofB(˜_n)in a similar fashion. We note that each strip is isomorphic to([0,2˜_n] × [0,2αlogn])∩Z². Takingαlarge enough, the following holds immediately by Lemma2.4and Lemma2.5.

Corollary 3.2. Fixd≥2andp > pc(Z^d).The following holdsPp-a.s.,for largen∈N:for all1≤i≤^d(d₂⁻¹⁾(2˜n+ 1)^d−2 and −_n≤m≤_n, there exist at least2c2.4.1αlogndisjoint horizontal (respectively,vertical)crossings consisting of white sites in themth horizontal(respectively,vertical)strip ofFi.

SetLn:= 2c2.4.1αlogn. Fix 1≤i≤ ^d(d₂⁻¹⁾(2˜n+1)^d−2and a configuration satisfying the event of Corol- lary 3.2. We can choose L_n disjoint crossings of white sites in each strip of F_i. Since F_i is isomorphic to [− ˜_n,˜_n]²∩Z², the horizontal and the vertical crossings intersect (see Figure 2) and form a grid onF_i. We will call it a Kesten grid. Note that Kesten grids onF_iandF_j do not intersect ifi=j in general.

Fig. 2. An illustration of disjoint crossings of white sites (thin solid lines) in strips (rectangles with dotted boundaries) ofF₁(square with thick solid boundary). These horizontal and vertical crossings intersect sinceF₁is isomorphic to[− ˜n,˜n]²∩Z².

3.3. Effective resistance estimate

In this subsection we prove Theorem1.1. To prove it, we only need to estimate the upper bound of effective resistances for pairs of vertices onW (F_i)∩Cⁿ,1≤i≤ ^d(d₂⁻¹⁾(2˜_n+1)^d−2 (recall the notations (3.2)–(3.5)). In the following proposition, we will focus our attention to W (F₁)∩Cⁿ. Recall the constant α in Section 3.2. To k₁, k₂∈Z, we associate the “square” defined by

S_kⁿ

1,k2:=

s₁e₁+s₂e₂: s_i−

2αlogn +1

k_i ≤ αlogn, i=1,2 .

Proposition 3.3. Fixd≥2andp > pc(Z^d).Letm1, m2,m˜2be integers satisfying thatm2<m˜2,m˜2−m2is even, andm₁+^m˜2−₂^m2 ≤_n.Then there existsc₁>0 (not depending on choices ofm₁, m₂,m˜₂)such thatPp-a.s.,for large n∈N,the following holds:for allx∈W (S_mⁿ

1,m₂)∩Cⁿandy∈W (S_mⁿ

1,m˜2)∩Cⁿ, R^C_effⁿ(x, y)≤c1·logn.

The upper bound of Theorem1.1follows from a similar argument of the proof of Proposition3.3below together with Lemma2.3and the triangle inequality for the effective resistance (see, for instance, [26], Corollary 10.8). We omit the details. The lower bound of Theorem 1.1holds by Lemma 2.6, (3.1) and the Nash–Williams inequality.

In the rest of the article, we focus on proving Proposition 3.3. Fix some integers r, r withr < r. A left-to-right (respectively, right-to-left) path in[r, r] ×Z^d−1is a path in[r, r] ×Z^d−1whose initial and final verticesa, bsatisfy a·e₁=randb·e₁=r(respectively,a·e₁=randb·e₁=r). A bottom-to-top path inZ× [r, r] ×Z^d−2is a path inZ× [r, r] ×Z^d−2whose initial and final verticesa, bsatisfya·e₂=randb·e₂=r. For a pathanda, a∈ (we assume thataappears beforeain), we write[a, a]to denote a part offromatoa. In particular, whena is the final vertex of, we write[a]in the place of[a, a].

Proof of Proposition3.3. We fix a configuration satisfying the events of Remark2.2, Corollary3.2, and Lemma2.3 withκlarge enough. We choose verticesxandyas in Proposition3.3. We will construct a unit flow betweenxandy. Our argument is 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 fromx toy, most of whose parts lie onW (F₁)∩Cⁿ. Recall the notations (3.6) and (3.7). A Kesten grid guarantees the existence of the following self-avoiding paths:

• disjoint left-to-right (respectively, right-to-left) self-avoiding paths of white sites H₁^m, . . . , H_L^m

n contained in Rⁿ₁(m2+m)for 1≤m≤^m˜2−₂^m2 (respectively, ^m˜2−₂^m2 < m <m˜2−m2),

• disjoint bottom-to-top self-avoiding paths of white sitesV₁^m, . . . , V_L^m

ncontained inRⁿ₂(m₁+m)for 0≤m≤^m˜2−₂^m2,

• left-to-right self-avoiding paths of white sites H_x andH_y of length at mostclognfor somec >0, contained in S_mⁿ

1,m₂ and inS_mⁿ

1,m˜2 respectively.

From now, we construct an open path fromxtoy corresponding to a fixed label(i, j )withi∈ {0, . . . ,^m˜2−₂^m2}and j ∈ {1, . . . , Ln}by the following three steps.

Step 1: For convenience, we will associate a vertex(k1, k2)∈Z² toS_kⁿ

1,k₂. Letf:{0, . . . ,2i} → {0, . . . , i}be a function defined byf (r)=rfor 0≤r≤iandf (r)=2i−rfori < r≤2i. Take a self-avoiding pathi as follows:

• Let (s_r)_0≤r≤2i+1 be a sequence with 0=s₀< s₁<· · ·< s_2i+1= ˜m₂−m₂. For 0≤r≤2i, set v_2r =(m₁+ f (r), m₂+s_r) andv_2r+1=(m₁+f (r), m₂+s_r+1). The self-avoiding path _i is obtained from the sequence (v_r)_0≤r≤4i+1by linear interpolation.

• The path_i lies within 1 in^∞-distance from the piecewise line segments connecting(m₁, m₂), (m₁+i,^{m2+ ˜}₂^m2) and(m₁,m˜₂).

See the left-hand side of Figure3.

Step 2: Recall the notation before the proof of Proposition3.3. Leth_x(respectively,h_y) be the initial (respectively, the final) vertex ofH_x(respectively,H_y). We define verticesa₁, . . . , a_4i+2as follows:

• For 0≤r≤2i,a2r+1is the last-visited vertex ofH_j^sr[a2r]byV_j^{f (r)}, wherea0:=hxandH_j^s0:=Hx.

Fig. 3. Illustrations of paths_i, (i, j )andγ (i, j ).

Fig. 4. Illustrations ofa_2r+1anda_2r+2for 0≤r≤2i.

• For 0≤r≤2i,a_2r+2is the last-visited vertex ofV_j^{f (r)}[a_2r+1]byH_j^sr+1, whereH_j^s2i+1:=H_y. See Figure4. We can obtain a self-avoiding path(i, j )of white sites fromh_xtoh_yin

(k1,k2)∈iS_kⁿ

1,k₂by connecting segments

H_j^sr[a2r, a2r+1], V_j^{f (r)}[a2r+1, a2r+2], 0≤r≤2i, and Hy[a4i+2, hy]. See the middle of Figure3.

Step 3: Recall the notation (2.1). Fix verticesx˜∈Bh_x(K)∩Cⁿandy˜∈Bh_y(K)∩Cⁿ. Indeed, we can take such

˜

x andy˜by Remark2.2. By the definition of white sites, we can take a self-avoiding open pathγ (i, j )˜ fromx˜ toy˜ contained inW ((i, j )). By Lemma2.3, we have self-avoiding open pathsγx andγy inCⁿ fromx tox˜ and fromy˜ toy respectively of length at mostclognfor somec >0. We can get a self-avoiding open pathγ (i, j )fromxtoy in Cⁿviaγx,γ (i, j )˜ andγy. See the right-hand side of Figure3.

LetX andY be random variables distributed uniformly on{0, . . . ,^m˜2−₂^m2}and on{1, . . . , Ln}respectively; they are defined on a probability space with probability measureP. From now, we construct a unit flow fromxtoyvia the random open pathγ (X, Y ). Recall the notation (1.1). We define a random functionψonCⁿ×Cⁿby

ψ (u, v):=

1 if{u, v} ∈O_Cⁿandγ (X, Y )passesu, vin this order,

−1 if{u, v} ∈O_Cⁿandγ (X, Y )passesv, uin this order, 0 otherwise.

We define a functionθonCⁿ×Cⁿby θ (u, v):=E

ψ (u, v)

foru, v∈Cⁿ.

Sinceψis a unit flow fromxtoy,θis a unit flow fromxtoy. In order to boundθ, let us define the following function onCⁿ×Cⁿ:

p(u, v):=

P

γ (X, Y )passes the edge{u, v}

if{u, v} ∈O_Cⁿ,

0 otherwise.

For 0≤r≤ ˜m₂−m₂, we define a set of labels of therth level by

D_r:= (m₁+s, m₂+r)∈Z²: 0≤s≤r+1

ifr≤^m˜2−₂^m2, (m₁+s, m₂+r)∈Z²: 0≤s≤ ˜m₂−m₂−r+1

otherwise.

LetU be the union ofγ_x,γ_y,W (H_x)andW (H_y). By Thomson’s principle (see, for example, [26], Theorem 9.10) and the construction ofγ (X, Y ), we have

R^C_effⁿ(x, y)≤1 2

u,v∈Cⁿ

θ (u, v)²

≤

u,v∈U

p(u, v)²+

(m˜₂−m₂)/2 r=0

(k1,k2)∈Dr

u,v∈W (Sⁿ_k

1,k2) v /∈U

p(u, v)²

+

˜ m2−m2

r=(m˜₂−m₂)/2+1

(k1,k2)∈Dr

u,v∈W (S_kⁿ

1,k2) v /∈U

p(u, v)². (3.8)

Since γ_x, γ_y, H_x andH_y are self-avoiding paths of length of order logn, the first term on the right-hand side of (3.8) is bounded byclognfor somec >0. Fix 1≤r≤ ^m˜2−₂^m2, (k1, k2)∈Dr andu, v∈W (Sⁿ_k

1,k₂)withv /∈U and {u, v} ∈O_Cⁿ. By the construction of the random open pathγ (X, Y ), we have for somec >0,

p(u, v)≤ c r·L_n.

So we have for somec, c>0,

(m˜2−m2)/2 r=0

(k₁,k₂)∈D_r

u,v∈W (S_kⁿ

1,k2) v /∈U

p(u, v)²≤c

(m˜2−m2)/2 r=1

1

r ≤clogn.

By a similar argument, the last term on the right-hand side of (3.8) is bounded byclognfor somec >0. Therefore

by (3.8), we obtain the conclusion.

Acknowledgements

The author would like to thank Professor Kumagai for variable comments on the early version of the manuscript and for suggesting that results in [13] are useful to prove Theorem1.1. The author would like to heartily thank Professor Kozma for a lot of helpful suggestions on the manuscript which lead to improvement in descriptions of the paper.

The author is deeply indebted to Dr. Gurel-Gurevich and Dr. Kagan for pointing out that the effective resistance for Cⁿshould be of order logn. The author would like to thank Dr. Fukushima and Dr. Shiraishi for encouragement and helpful comments. The author would like to thank the referee for careful reading of the article and useful comments.