Concentration estimates for the isoperimetric constant of the supercritical percolation cluster

Eviatar B. Procaccia (Weizmann Institute of Science)
Ron Rosenthal (The Hebrew University of Jerusalem)

Abstract


We consider the Cheeger constant $\phi(n)$ of the giant component of supercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show that the variance of $\phi(n)$ is bounded by $\frac{\xi}{n^d}$, where $\xi$ is a positive constant that depends only on the dimension $d$ and the percolation parameter.

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

Pages: 1-11

Publication Date: July 23, 2012

DOI: 10.1214/ECP.v17-2185

References

  • Alon, Noga; Krivelevich, Michael; Vu, Van H. On the concentration of eigenvalues of random symmetric matrices. Israel J. Math. 131 (2002), 259--267. MR1942311
  • Antal, Peter; Pisztora, Agoston. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), no. 2, 1036--1048. MR1404543
  • Berger, N.; Biskup, M.; Hoffman, C. E.; Kozma, G. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri PoincarĂ© Probab. Stat. 44 (2008), no. 2, 374--392. MR2446329
  • Benjamini, Itai; Kalai, Gil; Schramm, Oded. First passage percolation has sublinear distance variance. Ann. Probab. 31 (2003), no. 4, 1970--1978. MR2016607
  • Benjamini, Itai; Mossel, Elchanan. On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 (2003), no. 3, 408--420. MR1967022
  • Deuschel, Jean-Dominique; Pisztora, Agoston. Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 (1996), no. 4, 467--482. MR1384041
  • Mathieu, Pierre; Remy, Elisabeth. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004), no. 1A, 100--128. MR2040777
  • Pete, Gábor. A note on percolation on $\Bbb Z^ d$: isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab. 13 (2008), 377--392. MR2415145
  • E.B. Procaccia and E. Shellef. On the range of a random walk in a torus and random interlacements. Arxiv preprint arXiv:1007.1401, 2010.
  • Talagrand, Michel. On Russo's approximate zero-one law. Ann. Probab. 22 (1994), no. 3, 1576--1587. MR1303654
  • A. Timar. Bondary-connectivity via graph theory. To appear in Proceedings of the American Mathematical Scociety, 2007.
Bookmark and Share


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