The effect of small quenched noise on connectivity properties of random interlacements

Balázs Ráth (The University of British Columbia)
Artëm Sapozhnikov (Max-Planck Institute for Mathematics in the Sciences)


Random interlacements (at level $u$) is a one parameter family of random subsets of $\mathbb{Z}^d$ introduced by Sznitman. The vacant set at level $u$ is the complement of the random interlacement at level $u$. While the random interlacement induces a connected subgraph of $\mathbb{Z}^d$ for all levels $u$, the vacant set has a non-trivial phase transition in $u$.

In this paper, we study the effect of small quenched noise on connectivity properties of the random interlacement and the vacant set. For a positive $\varepsilon$, we allow each vertex of the random interlacement (referred to as occupied) to become vacant, and each vertex of the vacant set to become occupied with probability $\varepsilon$, independently of the randomness of the interlacement, and independently for different vertices. We prove that for any $d\geq 3$ and $u>0$, almost surely, the perturbed random interlacement percolates for small enough noise parameter $\varepsilon$. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is transient. As for the vacant set, we show that for any $d\geq 3$, there is still a non trivial phase transition in $u$ when the noise parameter $\varepsilon$ is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when $\varepsilon\to 0$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-20

Publication Date: January 7, 2013

DOI: 10.1214/EJP.v18-2122


  • Campanino, M.; Russo, L. An upper bound on the critical percolation probability for the three-dimensional cubic lattice. Ann. Probab. 13 (1985), no. 2, 478--491. MR0781418
  • Chayes, J. T.; Chayes, L.; Fisher, Daniel S.; Spencer, T. Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57 (1986), no. 24, 2999--3002. MR0925751
  • Chayes, J. T.; Chayes, L.; Fisher, Daniel S.; Spencer, T. Correlation length bounds for disordered Ising ferromagnets. Comm. Math. Phys. 120 (1989), no. 3, 501--523. MR0981216
  • Černý, J.; Popov, S. On the internal distance in the interlacement set. Elect. J. of Probab. 17 (2012), article 29.
  • Deuschel, Jean-Dominique; Pisztora, Agoston. Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 (1996), no. 4, 467--482. MR1384041
  • Doyle, Peter G.; Snell, J. Laurie. Random walks and electric networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. xiv+159 pp. ISBN: 0-88385-024-9 MR0920811
  • Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm. Local percolative properties of the vacant set of random interlacements with small intensity. arXiv:1206.6635 (2012).
  • Dunford, N; Schwartz, J.T. Linear operators, Volume 1, Wiley-Interscience, New York, 1958.
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339
  • Halperin, B.I; Weinrib, A. Critical phenomena in systems with long-range-correlated quenched disorder. Phys. Rev. B 27(1) (1983), 413--427.
  • Lacoin, Hubert; Tykesson, Johan. On the easiest way to connect k points in the random interlacements process. arXiv:1206.4216 (2012).
  • Pemantle, Robin; Peres, Yuval. On which graphs are all random walks in random environments transient? Random discrete structures (Minneapolis, MN, 1993), 207--211, IMA Vol. Math. Appl., 76, Springer, New York, 1996. MR1395617
  • Procaccia, Eviatar B.; Shellef, Eric. On the range of a random walk in a torus and random interlacements. arXiv:1007.1401v2 (2012).
  • Procaccia, Eviatar B.; Tykesson, Johan. Geometry of the random interlacement. Electron. Commun. Probab. 16 (2011), 528--544. MR2836759
  • Ráth, Balázs; Sapozhnikov, Artëm. On the transience of random interlacements. Electron. Commun. Probab. 16 (2011), 379--391. MR2819660
  • Ráth, Balázs; Sapozhnikov, Artëm. Connectivity properties of random interlacement and intersection of random walks. ALEA 9 (2012), 67--83.
  • Ráth, Balázs; Sapozhnikov, Artëm. On the effect of small quenched nosie on connectivity properties of random interlacements. arXiv:1109.5086v1 (2011).
  • Sidoravicius, Vladas; Sznitman, Alain-Sol. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 (2009), no. 6, 831--858. MR2512613
  • Sznitman, Alain-Sol. Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 (2009), no. 1-2, 143--174. MR2520124
  • Sznitman, Alain-Sol. Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37 (2009), no. 5, 1715--1746. MR2561432
  • Sznitman, Alain-Sol. Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 (2010), no. 3, 2039--2087. MR2680403
  • Sznitman, Alain-Sol. Decoupling inequalities and interlacement percolation on $G\times{\Bbb Z}$. Invent. Math. 187 (2012), no. 3, 645--706. MR2891880
  • Teixeira, Augusto. On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab. 19 (2009), no. 1, 454--466. MR2498684
  • Teixeira, Augusto. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Related Fields 150 (2011), no. 3-4, 529--574. MR2824866
  • Teixeira, Augusto; Windisch, David. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 (2011), no. 12, 1599--1646. MR2838338
  • Tímár, Á. Boundary-connectivity via graph theory. To appear in Proc. Amer. Math. Soc. (2012).
  • Weinrib, Abel. Long-range correlated percolation. Phys. Rev. B (3) 29 (1984), no. 1, 387--395. MR0729982
  • Windisch, David. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008), 140--150. MR2386070

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.