A lower bound for disconnection by random interlacements

Xinyi Li (ETH Zurich)
Alain-Sol Sznitman (ETH Zurich)

Abstract


We consider the vacant set of random interlacements on $\mathbb{Z}^d$, with $d$ bigger or equal to 3, in the percolative regime. Motivated by the large deviation principles obtained in our recent work arXiv:1304.7477, we investigate the asymptotic behavior of the probability that a large body gets disconnected from infinity by the random interlacements. We derive an asymptotic lower bound, which brings into play tilted interlacements, and relates the problem to some of the large deviations of the occupation-time profile considered in arXiv:1304.7477.

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

Pages: 1-26

Publication Date: January 28, 2014

DOI: 10.1214/EJP.v19-3067

References

  • Belius, David. Gumbel fluctuations for cover times in the discrete torus. Probab. Theory Related Fields 157 (2013), no. 3-4, 635--689. MR3129800
  • Benjamini, Itai; Sznitman, Alain-Sol. Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 1, 133--172. MR2349899
  • van den Berg, M.; Bolthausen, E.; den Hollander, F. Moderate deviations for the volume of the Wiener sausage. Ann. of Math. (2) 153 (2001), no. 2, 355--406. MR1829754
  • Cerf, Raphaël. Large deviations for three dimensional supercritical percolation. Astérisque No. 267 (2000), vi+177 pp. MR1774341
  • Černý, Jiří; Teixeira, Augusto; Windisch, David. Giant vacant component left by a random walk in a random $d$-regular graph. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 4, 929--968. MR2884219
  • Demuth, Michael; van Casteren, Jan A. Stochastic spectral theory for selfadjoint Feller operators. A functional integration approach. Probability and its Applications. Birkhäuser Verlag, Basel, 2000. xii+463 pp. ISBN: 3-7643-5887-4 MR1772266
  • Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations. Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9 MR0997938
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  • Fukushima, Masatoshi; Oshima, Yoichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. Second revised and extended edition. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 2011. x+489 pp. ISBN: 978-3-11-021808-4 MR2778606
  • 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
  • Lawler, Gregory F. Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991. 219 pp. ISBN: 0-8176-3557-2 MR1117680
  • Lawler, Gregory F.; Limic, Vlada. Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2 MR2677157
  • X. Li and A.S. Sznitman. Large deviations for occupation time profiles of random interlacements. To appear in Probab. Theory Relat. Fields, also available at arXiv:1304.7477.
  • S. Popov and A. Teixeira. Soft local times and decoupling of random interlacements. To appear in J. Eur. Math. Soc., also available at arXiv:1212.1605.
  • Port, Sidney C.; Stone, Charles J. Brownian motion and classical potential theory. Probability and Mathematical Statistics. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xii+236 pp. ISBN: 0-12-561850-6 MR0492329
  • Resnick, Sidney I. Extreme values, regular variation, and point processes. Applied Probability. A Series of the Applied Probability Trust, 4. Springer-Verlag, New York, 1987. xii+320 pp. ISBN: 0-387-96481-9 MR0900810
  • Sidoravicius, Vladas; Sznitman, Alain-Sol. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 (2009), no. 6, 831--858. MR2512613
  • Sidoravicius, Vladas; Sznitman, Alain-Sol. Connectivity bounds for the vacant set of random interlacements. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 4, 976--990. MR2744881
  • Sznitman, Alain-Sol. On the domination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14 (2009), no. 56, 1670--1704. MR2525107
  • 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
  • Sznitman, Alain-Sol. An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17 (2012), no. 9, 9 pp. MR2892408
  • Teixeira, Augusto; Windisch, David. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 (2011), no. 12, 1599--1646. MR2838338
Bookmark and Share


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