×

Supercritical loop percolation on \(\mathbb{Z}^d\) for \(d \geq 3\). (English) Zbl 1372.60132

Summary: We consider supercritical percolation on \(\mathbb{Z}^d\) \((d \geq 3)\) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all \(d \geq 3\) and all supercritical intensities despite polynomial decay of correlations.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
82B43 Percolation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aizenman, M.; Barsky, D. J., Sharpness of the phase transition in percolation models, Comm. Math. Phys., 108, 3, 489-526 (1987) · Zbl 0618.60098
[2] Aizenman, M.; Grimmett, G., Strict monotonicity for critical points in percolation and ferromagnetic models, J. Stat. Phys., 63, 5-6, 817-835 (1991)
[3] Antal, P.; Pisztora, A., On the chemical distance for supercritical bernoulli percolation, Ann. Probab., 24, 2, 1036-1048 (1996) · Zbl 0871.60089
[4] Barlow, M. T., Random walks on supercritical percolation clusters, Ann. Probab., 32, 4, 3024-3084 (2004) · Zbl 1067.60101
[5] Barlow, M. T.; Hambly, B. M., Parabolic Harnack inequality and local limit theorem for percolation clusters, Electron. J. Probab., 14, 1, 1-27 (2009) · Zbl 1192.60107
[6] Benjamini, I.; Duminil-Copin, H.; Kozma, G.; Yadin, A., Disorder, entropy and harmonic functions, Ann. Probab., 43, 5, 2332-2373 (2015) · Zbl 1337.60248
[7] Berger, N., Transience, recurrence and critical behavior for long-range percolation, Comm. Math. Phys., 226, 3, 531-558 (2002) · Zbl 0991.82017
[8] Berger, N.; Biskup, M., Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields, 137, 1-2, 83-120 (2007) · Zbl 1107.60066
[9] Chang, Y.; Sapozhnikov, A., Phase transition in loop percolation, Probab. Theory Related Fields, 164, 3-4, 979-1025 (2016) · Zbl 1341.60123
[10] Chayes, J. T.; Chayes, L.; Newman, C. M., Bernoulli percolation above threshold: an invasion percolation analysis, Ann. Probab., 15, 4, 1272-1287 (1987) · Zbl 0627.60099
[11] Croydon, D. A.; Hambly, B. M., Local limit theorems for sequences of simple random walks on graphs, Potential Anal., 29, 4, 351-389 (2008) · Zbl 1185.60087
[12] Drewitz, A.; Ráth, B.; Sapozhnikov, A., Local percolative properties of the vacant set of random interlacements with small intensity, Ann. Inst. Henri Poincaré Probab. Stat., 50, 4, 1165-1197 (2014) · Zbl 1319.60180
[13] Drewitz, A.; Ráth, B.; Sapozhnikov, A., On chemical distances and shape theorems in percolation models with long-range correlations, J. Math. Phys., 55, 8, Article 083307 pp. (2014), 30 · Zbl 1301.82027
[14] Grimmett, G., (Percolation. Percolation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321 (1999), Springer-Verlag: Springer-Verlag Berlin), xiv+444
[15] Grimmett, G. R.; Marstrand, J. M., The supercritical phase of percolation is well behaved, Proc. R. Soc. Lond. Ser. A, 430, 1879, 439-457 (1990) · Zbl 0711.60100
[16] Hopf, E., An inequality for positive linear integral operators, J. Math. Mech., 12, 683-692 (1963) · Zbl 0115.32501
[17] Lawler, G. F.; Limic, V., Random walk: a modern introduction, (Cambridge Studies in Advanced Mathematics, vol. 123 (2010), Cambridge University Press: Cambridge University Press Cambridge), xii+364 · Zbl 1210.60002
[18] Lawler, G. F.; Trujillo Ferreras, J. A., Random walk loop soup, Trans. Amer. Math. Soc., 359, 2, 767-787 (2007), (electronic) http://dx.doi.org/10.1090/S0002-9947-06-03916-X · Zbl 1120.60037
[19] Lawler, G. F.; Werner, W., The Brownian loop soup, Probab. Theory Related Fields, 128, 4, 565-588 (2004) · Zbl 1049.60072
[20] Le Jan, Y., Markov paths, loops and fields, (Lecture Notes in Mathematics, vol. 2026 (2011), Springer: Springer Heidelberg), viii+124, Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]
[21] Le Jan, Y.; Lemaire, S., Markovian loop clusters on graphs, Illinois J. Math., 57, 2, 525-558 (2013), URL http://projecteuclid.org/euclid.ijm/1408453593 · Zbl 1311.60015
[23] Lupu, T., From loop clusters and random interlacements to the free field, Ann. Probab., 44, 3, 2117-2146 (2016) · Zbl 1348.60141
[24] Lupu, T., Loop percolation on discrete half-plane, Electron. Commun. Probab., 21 (2016), Paper No. 30, 9 · Zbl 1338.60235
[25] Meester, R.; Steif, J. E., On the continuity of the critical value for long range percolation in the exponential case, Comm. Math. Phys., 180, 2, 483-504 (1996) · Zbl 0878.60068
[26] Popov, S.; Ráth, B., On decoupling inequalities and percolation of excursion sets of the gaussian free field, J. Stat. Phys., 159, 2, 312-320 (2015) · Zbl 1328.82026
[27] Popov, S.; Teixeira, A., Soft local times and decoupling of random interlacements, J. Eur. Math. Soc. (JEMS), 17, 10, 2545-2593 (2015) · Zbl 1329.60342
[28] Procaccia, E. B.; Rosenthal, R.; Sapozhnikov, A., Quenched invariance principle for simple random walk on clusters in correlated percolation models, Probab. Theory Related Fields, 166, 3-4, 619-657 (2016) · Zbl 1353.60034
[29] Rodriguez, P.-F.; Sznitman, A.-S., Phase transition and level-set percolation for the gaussian free field, Comm. Math. Phys., 320, 2, 571-601 (2013) · Zbl 1269.82028
[30] Sapozhnikov, A., Random walks on infinite percolation clusters in models with long-range correlations, Ann. Probab. (2014), in press URL http://arxiv.org/abs/1410.0605
[31] Sheffield, S.; Werner, W., Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. of Math. (2), 176, 3, 1827-1917 (2012) · Zbl 1271.60090
[32] Sznitman, A.-S., Vacant set of random interlacements and percolation, Ann. of Math. (2), 171, 3, 2039-2087 (2010) · Zbl 1202.60160
[33] Sznitman, A.-S., Decoupling inequalities and interlacement percolation on \(G \times Z\), Invent. Math., 187, 3, 645-706 (2012) · Zbl 1277.60183
[34] Tassion, V., Crossing probabilities for Voronoi percolation, Ann. Probab., 44, 5, 3385-3398 (2016) · Zbl 1352.60130
[35] Teixeira, A., On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Related Fields, 150, 3-4, 529-574 (2011) · Zbl 1231.60117
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.