Random interlacements and amenability. (English) Zbl 1375.60139
Summary: We consider the model of random interlacements on transient graphs, which was first introduced by A.-S. Sznitman [Ann. Math. (2) 171, No. 3, 2039–2087 (2010; Zbl 1202.60160)] for the special case of \({\mathbb{Z}}^{d}\) (with \(d\geq 3\)). In [loc. cit.], it was shown that on \({\mathbb{Z}}^{d}\): for any intensity \(u>0\), the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity \(u\) the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of \(u\). Finally, we establish the monotonicity of the transition between the “disconnected” and the “connected” phases, providing the uniqueness of the critical value \(u_{c}\) where this transition occurs.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
Citations:
Zbl 1202.60160References:
| [1] | Alves, O. S. M., Machado, F. P. and Popov, S. Y. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533-546. · Zbl 1013.60081 · doi:10.1214/aoap/1026915614 |
| [2] | Belius, D. (2012). Cover levels and random interlacements. Ann. Appl. Probab. 22 522-540. · Zbl 1271.60057 · doi:10.1214/11-AAP770 |
| [3] | Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29-66. · Zbl 0924.43002 · doi:10.1007/s000390050080 |
| [4] | Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1-65. · Zbl 1016.60009 |
| [5] | Benjamini, I. and Schramm, O. (1996). Percolation beyond \(Z^{d}\), many questions and a few answers. Electron. Commun. Probab. 1 71-82 (electronic). · Zbl 0890.60091 · doi:10.1214/ECP.v1-978 |
| [6] | Benjamini, I. and Schramm, O. (2001). Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 487-507 (electronic). · Zbl 1037.82018 · doi:10.1090/S0894-0347-00-00362-3 |
| [7] | Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505. · Zbl 0662.60113 · doi:10.1007/BF01217735 |
| [8] | Dodziuk, J. (1984). Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 787-794. · Zbl 0512.39001 · doi:10.2307/1999107 |
| [9] | Grimmett, G. R. and Newman, C. M. (1990). Percolation in \(\infty +1\) dimensions. In Disorder in Physical Systems 167-190. Oxford Univ. Press, New York. · Zbl 0721.60121 |
| [10] | Häggström, O. and Jonasson, J. (2006). Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 289-344. · Zbl 1189.60175 · doi:10.1214/154957806000000096 |
| [11] | Häggström, O. and Peres, Y. (1999). Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273-285. · Zbl 0921.60091 · doi:10.1007/s004400050208 |
| [12] | Pak, I. and Smirnova-Nagnibeda, T. (2000). On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330 495-500. · Zbl 0947.43003 · doi:10.1016/S0764-4442(00)00211-1 |
| [13] | Ramírez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. ( JEMS ) 6 293-334. · Zbl 1049.60089 · doi:10.4171/JEMS/11 |
| [14] | Resnick, S. I. (2008). Extreme Values , Regular Variation and Point Processes . Springer, New York. Reprint of the 1987 original. · Zbl 1136.60004 |
| [15] | Schonmann, R. H. (1999). Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields 113 287-300. · Zbl 0921.60092 · doi:10.1007/s004400050209 |
| [16] | Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831-858. · Zbl 1168.60036 · doi:10.1002/cpa.20267 |
| [17] | Sznitman, A.-S. (2008). How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36 1-53. · Zbl 1134.60061 · doi:10.1214/009117907000000114 |
| [18] | Sznitman, A.-S. (2009). On the domination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14 1670-1704. · Zbl 1196.60170 · doi:10.1214/EJP.v14-679 |
| [19] | Sznitman, A.-S. (2009). Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 143-174. · Zbl 1172.60316 · doi:10.1007/s00440-008-0164-8 |
| [20] | Sznitman, A.-S. (2009). Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37 1715-1746. · Zbl 1179.60025 · doi:10.1214/09-AOP450 |
| [21] | Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039-2087. · Zbl 1202.60160 · doi:10.4007/annals.2010.171.2039 |
| [22] | Sznitman, A.-S. (2012). Decoupling inequalities and interlacement percolation on \(G\times\mathbb{Z}\). Invent. Math. 187 645-706. · Zbl 1277.60183 · doi:10.1007/s00222-011-0340-9 |
| [23] | Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 1604-1628. · Zbl 1192.60108 · doi:10.1214/EJP.v14-670 |
| [24] | Teixeira, A. and Windisch, D. (2011). On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 1599-1646. · Zbl 1235.60143 · doi:10.1002/cpa.20382 |
| [25] | Černý, J., Teixeira, A. and Windisch, D. (2011). Giant vacant component left by a random walk in a random \(d\)-regular graph. Ann. Inst. Henri Poincaré Probab. Stat. 47 929-968. · Zbl 1267.05237 · doi:10.1214/10-AIHP407 |
| [26] | Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140-150. · Zbl 1187.60089 · doi:10.1214/ECP.v13-1359 |
| [27] | Windisch, D. (2010). Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38 841-895. · Zbl 1191.60062 · doi:10.1214/09-AOP497 |
| [28] | Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138 . Cambridge Univ. Press, Cambridge. · Zbl 0951.60002 · doi:10.1017/CBO9780511470967 |
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