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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.

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
Full Text: DOI
[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)
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