Effective resistances for supercritical percolation clusters in boxes

Abe, Yoshihiro

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 935-946.

On considère la percolation de Bernoulli par arêtes dans le régime surcritique. Soit ${𝒞}^n$ le plus grand amas de percolation dans ${[-n,n]}^d\cap {\mathbb{Z}}^d$ avec $d\ge 2$. Nous obtenons une estimation précise de la résistance effective sur ${𝒞}^n$. Comme application, nous montrons que le temps de recouvrement d’une marche simple sur ${𝒞}^n$ est de l’ordre de $n^d{(logn)}^2$. En remarquant que le temps de recouvrement d’une marche simple sur ${[-n,n]}^d\cap {\mathbb{Z}}^d$ est de l’ordre de $n^{d}logn$ quand $d\ge 3$ (et de $n^2{(logn)}^2$ quand $d=2$), ceci montre une différence quantitative entre les deux marches si $d\ge 3$.

Let ${𝒞}^n$ be the largest open cluster for supercritical Bernoulli bond percolation in ${[-n,n]}^d\cap {\mathbb{Z}}^d$ with $d\ge 2$. We obtain a sharp estimate for the effective resistance on ${𝒞}^n$. As an application we show that the cover time for the simple random walk on ${𝒞}^n$ is comparable to $n^d{(logn)}^2$. Noting that the cover time for the simple random walk on ${[-n,n]}^d\cap {\mathbb{Z}}^d$ is of order $n^{d}logn$ for $d\ge 3$ (and of order $n^2{(logn)}^2$ for $d=2$), this gives a quantitative difference between the two random walks for $d\ge 3$.

Mots clés : effective resistances, simple random walks, cover times, gaussian free fields, supercritical percolation

[1] Y. Abe. Cover times for sequences of reversible Markov chains on random graphs. Kyoto J. Math. 54 (2014) 555–576. | MR | Zbl

[2] O. Angel, I. Benjamini, N. Berger and Y. Peres. Transience of percolation clusters on wedges. Electron. J. Probab. 11 (2006) 655–669. | EuDML | MR | Zbl

[3] P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036–1048. | DOI | MR | Zbl

[4] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004) 3024–3084. | DOI | MR | Zbl

[5] M. T. Barlow, Y. Peres and P. Sousi. Collisions of random walks. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 922–946. | DOI | EuDML | Numdam | MR | Zbl

[6] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007) 83–120. | DOI | MR | Zbl

[7] I. Benjamini and G. Kozma. A resistance bound via an isoperimetric inequality. Combinatorica 25 (2005) 645–650. Available at arXiv:math/0212322v2. | DOI | MR | Zbl

[8] I. Benjamini and E. Mossel. On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 (2003) 408–420. | DOI | MR | Zbl

[9] I. Benjamini, R. Pemantle and Y. Peres. Unpredictable paths and percolation. Ann. Probab. 26 (1998) 1198–1211. | DOI | MR | Zbl

[10] D. Boivin and C. Rau. Existence of the harmonic measure for random walks on graphs and in random environments. J. Stat. Phys. 150 (2013) 235–263. Available at arXiv:1111.5326v2. | DOI | MR | Zbl

[11] J. T. Chayes and L. Chayes. Bulk transport properties and exponent inequalities for random resistor and flow networks. Comm. Math. Phys. 105 (1986) 133–152. | DOI | MR | Zbl

[12] X. Chen and D. Chen. Two random walks on the open cluster of ${\mathbb{Z}}^2$ meet infinitely often. Sci. China Math. 53 (2010) 1971-1978. | DOI | MR | Zbl

[13] Z.-Q. Chen, D. A. Croydon and T. Kumagai. Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. Available at arXiv:1306.0076v1. | MR | Zbl

[14] O. Couronné and R. J. Messikh. Surface order large deviations for 2D FK-percolation and Potts models. Stochastic Process. Appl. 113 (2004) 81–99. | DOI | MR | Zbl

[15] A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky and P. Tiwari. The electrical resistance of a graph captures its commute and cover times. Comput. Complexity 6 (1996/1997) 312–340. | MR | Zbl

[16] J. Ding, J. R. Lee and Y. Peres. Cover times, blanket times, and majorizing measures. Ann. of Math. (2) 175 (2012) 1409–1471. | MR | Zbl

[17] P. G. Doyle and J. L. Snell. Random Walks and Electric Networks. Carus Math. Monogr. 22. Math. Assoc. Amer., Washington, DC, 1984. | MR | Zbl

[18] G. Grimmett and H. Kesten. First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 (1984) 335–366. | DOI | MR | Zbl

[19] G. R. Grimmett, H. Kesten and Y. Zhang. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993) 33–44. | DOI | MR | Zbl

[20] O. Häggström and E. Mossel. Nearest-neighbor walks with low predictability profile and percolation in $2+\epsilon$ dimensions. Ann. Probab. 26 (1998) 1212–1231. | DOI | MR | Zbl

[21] C. Hoffman. Energy of flows on $Z^2$ percolation clusters. Random Structures Algorithms 16 (2000) 143–155. | DOI | MR | Zbl

[22] C. Hoffman and E. Mossel. Energy of flows on percolation clusters. Potential Anal. 14 (2001) 375–385. | DOI | MR | Zbl

[23] H. Kesten. Percolation Theory for Mathematicians. Progress in Probability and Statistics. Birkhäuser, Boston, MA, 1982. | DOI | MR | Zbl

[24] G. Kozma. Personal communication.

[25] T. Kumagai. Random Walks on Disordered Media and Their Scaling Limits. École d’Été de Probabilités de Saint-Flour XL-2010. Lecture Notes in Mathematics 2101. Springer, Cham, 2014. | MR | Zbl

[26] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. | MR | Zbl

[27] T. M. Liggett, R. H. Schonmann and A. M. Stacey. Domination by product measures. Ann. Probab. 25 (1997) 71–95. | DOI | MR | Zbl

[28] D. Levin and Y. Peres. Energy and cutsets in infinite percolation clusters. In Proceedings of the Cortona Workshop on Random Walks and Discrete Potential Theory 264–278. M. Picardello and W. Woess (Eds). Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[29] R. Lyons and Y. Peres. Probability on Trees and Networks. Book in preparation. Current version available at http://mypage.iu.edu/~rdlyons/. | DOI | Zbl

[30] P. Mathieu and A. L. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007) 2287–2307. | MR | Zbl

[31] P. Mathieu and E. Remy. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004) 100–128. | DOI | MR | Zbl

[32] P. Matthews. Covering problems for Brownian motion on spheres. Ann. Probab. 16 (1988) 189–199. | DOI | MR | Zbl

[33] G. Pete. A note on percolation on ${\mathbb{Z}}^d$: Isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab. 13 (2008) 377–392. | DOI | MR | Zbl

[34] A. Pisztora. Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 (1996) 427–466. | MR | Zbl

[35] V. Sidoravicius and A.-S. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004) 219–244. | DOI | MR | Zbl