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Domination by product measures. (English) Zbl 0882.60046
Summary: We consider families of \(\{0,1\}\)-valued random variables indexed by the vertices of countable graphs with bounded degree. First we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value \(p\) which is large enough, then this random field dominates a product measure with positive density. Moreover the density of this dominated product measure can be made arbitrarily close to 1, provided that \(p\) is close enough to 1. Next we address the issue of obtaining the critical values of \(p\), defined as the threshold above which the domination by positive-density product measures is assured. For the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than \(k\) units, this critical value is shown to be \(1-k^k/(k+1)^{k+1}\), and a discontinuous transition is shown to occur. Similar critical values of \(p\) are found for other classes of probability measures on \(\{0,1\}^\mathbb{Z}\). For the class of \(k\)-dependent measures the critical value is again \(1-k^k/(k +1)^{k+1}\), with a discontinuous transition. For the class of two-block factors the ctitical value is shown to be 1/2 and a continuous transition is shown to take place in this case. Thus both the critical value and the nature of the transition are different in the two-block factor and 1-dependent cases.

MSC:
60G60 Random fields
60G10 Stationary stochastic processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] AARONSON, J., GILAT, D., KEANE, M. and DE VALK, V. 1989. An algebraic construction of a class of one-dependent processes. Ann. Probab. 17 128 143. · Zbl 0681.60038 · doi:10.1214/aop/1176991499
[2] ANDJEL, E. 1993. Characteristic exponents for two-dimensional bootstrap percolation. Ann. Probab. 21 926 935. · Zbl 0787.60120 · doi:10.1214/aop/1176989275
[3] ANTAL, P. and PISZTORA, A. 1996. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036 1048. · Zbl 0871.60089 · doi:10.1214/aop/1039639377
[4] BOLLOBAS, B. 1985. Random Graphs. Academic Press, London. \' · Zbl 0592.05052
[5] DURRETT, R. 1988. Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks Cole, Pacific Grove, CA. · Zbl 0659.60129
[6] ERDOS, P. and LOVASZ, L. 1975. Problems and results on 3-chromatic hypergraphs and some ŕelated results. In Infinite and Finite Sets A. Hajnal, R. Rado and V. T. Sos, eds. Coll. Ḿath. Soc. Janos Bolyai 11 609 627. Ź. · Zbl 0315.05117
[7] LIGGETT, T. M. 1985. Interacting Particle Systems. Springer, New York. · Zbl 0559.60078
[8] PENROSE, M. and PISZTORA, A. 1996. Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29 52. JSTOR: · Zbl 0853.60085 · doi:10.2307/1427912 · links.jstor.org
[9] PISZTORA, A. 1996. Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427 466. · Zbl 0842.60022 · doi:10.1007/BF01198161
[10] RUSSO, L. 1982. An approximate zero-one law.Wahrsch. Verw. Gebiete 61 129 139. Z. · Zbl 0501.60043 · doi:10.1007/BF00537230
[11] SCHONMANN, R. H. 1994. Theorems and conjectures on the droplet-driven relaxation of stochastic Ising models. In Probability and Phase Transition G. Grimmett, ed. 265 301. Kluwer, Dordrecht. Z. · Zbl 0835.60085
[12] SHEARER, J. B. 1985. On a problem of Spencer. Combinatorica 5 241 245. · Zbl 0587.60012 · doi:10.1007/BF02579368
[13] LOS ANGELES, CALIFORNIA 90024 E-MAIL: tml@math.ucla.edu rhs@math.ucla.edu stacey@math.ucla.edu
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