×

zbMATH — the first resource for mathematics

Surface order large deviations for high-density percolation. (English) Zbl 0842.60023
We derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a Bernoulli percolation process restricted to a big finite box, with sufficiently large parameter. We also establish a useful version of the isoperimetric inequality, which is the main tool of our proofs.

MSC:
60F10 Large deviations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Comm. Math. Phys.108, 489–526 (1987) · Zbl 0618.60098 · doi:10.1007/BF01212322
[2] Alexander, K., Chayes, J.T., Chayes, L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys.131, 1–50 (1990) · Zbl 0698.60098 · doi:10.1007/BF02097679
[3] Durrett, R., Schonmann, R.H.: Large deviations for the contact process and two dimensional percolation. Probab. Theory Relat. Fields77, 583–603 (1988) · Zbl 0621.60108 · doi:10.1007/BF00959619
[4] Fontes, L., Newman, C.M.: First-passage percolation for random colorings of 482-1. Ann. Appl. Probab.3, 746–762 (1993) · Zbl 0780.60101 · doi:10.1214/aoap/1177005361
[5] Gandolfi, A.: Clustering and uniqueness in mathematical models of percolation phenomena. Thesis (1989) · Zbl 0694.60096
[6] Grimmett, G.R.: Percolation. Berlin: Springer 1989
[7] Kesten, H.: Aspects of first passage percolation (Lect. Notes Math., Vol. 1180, pp. 125–264) Berlin: Springer 1986 · Zbl 0602.60098
[8] Kuratowski, K.: Topology II. New York: Academic Press 1968
[9] Menshikov, M.V.: Coincidence of critical points in percolation problems. Sov. Math. Dokl.33, 856–859 (1986) · Zbl 0615.60096
[10] Nagaev, S.V.: Large deviations of sums of independent random variables. Ann. Probab.7, 745–789 (1979) · Zbl 0418.60033 · doi:10.1214/aop/1176994938
[11] Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc.55, 961–962 (1949) · Zbl 0035.38302 · doi:10.1090/S0002-9904-1949-09320-5
[12] Pisztora, A.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Relat. Fields104, 427–466 (1996) · Zbl 0842.60022 · doi:10.1007/BF01198161
[13] Taylor, J.E.: Existence and structure of solutions to a class of nonelliptic variational problems. Symposia MathematicaXIV, 499–508 (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.