×

zbMATH — the first resource for mathematics

On chemical distances and shape theorems in percolation models with long-range correlations. (English) Zbl 1301.82027
Summary: In this paper, we provide general conditions on a one parameter family of random infinite subsets of \({\mathbb{Z}}^d\) to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of J. Černý and S. Popov [Electron. J. Probab. 17, Paper No. 29, 25 p. (2012; Zbl 1245.60090)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of P. Antal and A. Pisztora [Ann. Probab. 24, No. 2, 1036–1048 (1996; Zbl 0871.60089)] for Bernoulli percolation. Finally, as a corollary, we derive new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.
©2014 American Institute of Physics

MSC:
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alves, O. S. M.; Machado, F. P.; Popov, S. Yu., The shape theorem for the frog model, Ann. Appl. Probab., 12, 2, 533-546, (2002) · Zbl 1013.60081
[2] Antal, P.; Pisztora, A., On the chemical distance for supercritical Bernoulli percolation, Ann. Probab., 24, 2, 1036-1048, (1996) · Zbl 0871.60089
[3] Barlow, M. T., Random walks on supercritical percolation clusters, Ann. Probab., 32, 3024-3084, (2004) · Zbl 1067.60101
[4] Benjamini, I.; Sznitman, A.-S., Giant component and vacant set for random walk on a discrete torus, J. Eur. Math. Soc., 10, 1, 133-172, (2008) · Zbl 1141.60057
[5] Berger, N.; Biskup, M., Quenched invariance principle for simple random walk on percolation cluster, Probab. Theory Rel. Fields, 137, 83-120, (2007) · Zbl 1107.60066
[6] Bricmont, J.; Lebowitz, J. L.; Maes, C., Percolation in strongly correlated systems: The massless Gaussian field, J. Stat. Phys., 48, 5-6, 1249-1268, (1987) · Zbl 0962.82520
[7] Broadbent, S. R.; Hammersley, J. M., Percolation processes I. Crystals and mazes, Proc. Camb. Phil. Soc., 53, 629-641, (1957) · Zbl 0091.13901
[8] Černý, J.; Popov, S., On the internal distance in the interlacement set, Electron. J. Probab., 17, 29, 1-25, (2012) · Zbl 1245.60090
[9] Drewitz, A.; Ráth, B.; Sapozhnikov, A., Local percolative properties of the vacant set of random interlacements with small intensity, (2012) · Zbl 1319.60180
[10] Dunford, N.; Schwartz, J. T., Linear Operators, 1, (1958), Wiley-Interscience: Wiley-Interscience, New York
[11] Garet, O.; Marchand, R., Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster, ESAIM: Probab. Stat., 8, 169-199, (2004) · Zbl 1154.60356
[12] Gärtner, J.; Molchanov, S. A., Parabolic problems for the Anderson model, Commun. Math. Phys., 132, 613-655, (1990) · Zbl 0711.60055
[13] Grimmett, G. R.; Marstrand, J. M., The supercritical phase of percolation is well behaved, Proc. Roy. Soc. London Ser. A, 430, 439-457, (1990) · Zbl 0711.60100
[14] Grimmett, G. R., Percolation, (1999), Springer-Verlag: Springer-Verlag, Berlin
[15] Grimmett, G. R., The Random-Cluster Model, (2006), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1122.60087
[16] Kingman, J. F. C., Subadditive ergodic theory, Ann. Probab., 1, 883-909, (1973) · Zbl 0311.60018
[17] Lacoin, H.; Tykesson, J., On the easiest way to connect k points in the random interlacements process, ALEA, 10, 505-524, (2013) · Zbl 1277.60177
[18] Liggett, T. M.; Schonmann, R. H.; Stacey, A. M., Domination by product measures, Ann. Probab., 25, 71-95, (1997) · Zbl 0882.60046
[19] Mathieu, P.; Remy, E., Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab., 32, 100-128, (2004) · Zbl 1078.60085
[20] Mathieu, P.; Piatnitski, A. L., Quenched invariance principles for random walks on percolation clusters, Proc. Roy. Soc. A, 463, 2287-2307, (2007) · Zbl 1131.82012
[21] Okamura, K., Large deviations for simple random walk on percolations with long-range correlations, (2013)
[22] Popov, S.; Ráth, B., On decoupling inequalities and percolation of excursion sets of the Gaussian free field, (2013)
[23] Popov, S.; Teixeira, A., Soft local times and decoupling of random interlacements, J. Eur. Math. Soc., (2012)
[24] Procaccia, E.; Shellef, E., On the range of a random walk in a torus and random interlacements, Ann. Probab., (2012)
[25] Procaccia, E.; Tykesson, J., Geometry of the random interlacement, Electron. Commun. Probab., 16, 528-544, (2011) · Zbl 1254.60018
[26] Procaccia, E.; Rosenthal, R.; Sapozhnikov, A., Quenched invariance principle for simple random walk on clusters in correlated percolation models, (2013)
[27] Ráth, B.; Sapozhnikov, A., Connectivity properties of random interlacement and intersection of random walks, ALEA, 9, 67-83, (2012) · Zbl 1277.60182
[28] Ráth, B.; Sapozhnikov, A., On the transience of random interlacements, Electron. Commun. Probab., 16, 379-391, (2011) · Zbl 1231.60115
[29] Ráth, B.; Sapozhnikov, A., The effect of small quenched noise on connectivity properties of random interlacements, Electron. J. Prob., 18, 4, 1-20, (2011) · Zbl 1347.60132
[30] Rodriguez, P. F.; Sznitman, A.-S., Phase transition and level-set percolation for the Gaussian free field, Commun. Math. Phys., 320, 571-601, (2013) · Zbl 1269.82028
[31] Sidoravicius, V.; Sznitman, A.-S., Quenched invariance principles for walks on clusters of percolation or among random conductances, Prob. Theory Relat. Fields, 129, 219-244, (2004) · Zbl 1070.60090
[32] Sidoravicius, V.; Sznitman, A.-S., Percolation for the vacant set of random interlacements, Commun. Pure Appl. Math., 62, 6, 831-858, (2009) · Zbl 1168.60036
[33] Sznitman, A.-S., Vacant set of random interlacements and percolation, Ann. Math., 171, 2, 2039-2087, (2010) · Zbl 1202.60160
[34] Sznitman, A.-S., Decoupling inequalities and interlacement percolation on \documentclass[12pt]minimal\begindocument\(G× {\mathbb{Z}}\)\enddocument, Invent. Math., 187, 3, 645-706, (2012) · Zbl 1277.60183
[35] Teixeira, A., On the uniqueness of the infinite cluster of the vacant set of random interlacements, Ann. Appl. Probab., 19, 1, 454-466, (2009) · Zbl 1158.60046
[36] Teixeira, A., On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Relat. Fields, 150, 3-4, 529-574, (2011) · Zbl 1231.60117
[37] Teixeira, A.; Windisch, D., On the fragmentation of a torus by random walk, Commun. Pure Appl. Math., 64, 12, 1599-1646, (2011) · Zbl 1235.60143
[38] Windisch, D., Random walk on a discrete torus and random interlacements, Electron. Commun. Probab., 13, 140-150, (2008) · Zbl 1187.60089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.