## The Annals of Applied Probability

### Random interlacements and amenability

#### Abstract

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039–2087] for the special case of ${\mathbb{Z}}^{d}$ (with $d\geq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039–2087], 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.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 923-956.

Dates
First available in Project Euclid: 7 March 2013

https://projecteuclid.org/euclid.aoap/1362684850

Digital Object Identifier
doi:10.1214/12-AAP860

Mathematical Reviews number (MathSciNet)
MR3076674

Zentralblatt MATH identifier
1375.60139

#### Citation

Teixeira, Augusto; Tykesson, Johan. Random interlacements and amenability. Ann. Appl. Probab. 23 (2013), no. 3, 923--956. doi:10.1214/12-AAP860. https://projecteuclid.org/euclid.aoap/1362684850

#### References

• [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.
• [2] Belius, D. (2012). Cover levels and random interlacements. Ann. Appl. Probab. 22 522–540.
• [3] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66.
• [4] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1–65.
• [5] Benjamini, I. and Schramm, O. (1996). Percolation beyond $Z^{d}$, many questions and a few answers. Electron. Commun. Probab. 1 71–82 (electronic).
• [6] Benjamini, I. and Schramm, O. (2001). Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 487–507 (electronic).
• [7] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
• [8] Dodziuk, J. (1984). Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 787–794.
• [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.
• [10] Häggström, O. and Jonasson, J. (2006). Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 289–344.
• [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.
• [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.
• [13] Ramírez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. (JEMS) 6 293–334.
• [14] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York. Reprint of the 1987 original.
• [15] Schonmann, R. H. (1999). Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields 113 287–300.
• [16] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
• [17] Sznitman, A.-S. (2008). How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36 1–53.
• [18] Sznitman, A.-S. (2009). On the domination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14 1670–1704.
• [19] Sznitman, A.-S. (2009). Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 143–174.
• [20] Sznitman, A.-S. (2009). Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37 1715–1746.
• [21] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
• [22] Sznitman, A.-S. (2012). Decoupling inequalities and interlacement percolation on $G\times\mathbb{Z}$. Invent. Math. 187 645–706.
• [23] Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 1604–1628.
• [24] Teixeira, A. and Windisch, D. (2011). On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 1599–1646.
• [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.
• [26] Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140–150.
• [27] Windisch, D. (2010). Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38 841–895.
• [28] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.