The Annals of Probability

Domination by product measures

T. M. Liggett, R. H. Schonmann, and A. M. Stacey

Full-text: Open access


4 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 value 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 critical 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.

Article information

Ann. Probab., Volume 25, Number 1 (1997), 71-95.

First available in Project Euclid: 18 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 60G10: Stationary processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stochastic domination random fields product measures critical points rescaling


Liggett, T. M.; Schonmann, R. H.; Stacey, A. M. Domination by product measures. Ann. Probab. 25 (1997), no. 1, 71--95. doi:10.1214/aop/1024404279.

Export citation


  • 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.
  • ANDJEL, E. 1993. Characteristic exponents for two-dimensional bootstrap percolation. Ann. Probab. 21 926 935.
  • ANTAL, P. and PISZTORA, A. 1996. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036 1048.
  • BOLLOBAS, B. 1985. Random Graphs. Academic Press, London. ´
  • DURRETT, R. 1988. Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks Cole, Pacific Grove, CA.
  • ERDOS, P. and LOVASZ, L. 1975. Problems and results on 3-chromatic hypergraphs and some ´related results. In Infinite and Finite Sets A. Hajnal, R. Rado and V. T. Sos, eds. Coll. ´ Math. Soc. Janos Bolyai 11 609 627. ´ Z.
  • LIGGETT, T. M. 1985. Interacting Particle Systems. Springer, New York.
  • PENROSE, M. and PISZTORA, A. 1996. Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29 52.
  • PISZTORA, A. 1996. Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427 466.
  • RUSSO, L. 1982. An approximate zero-one law.Wahrsch. Verw. Gebiete 61 129 139. Z.
  • 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.
  • SHEARER, J. B. 1985. On a problem of Spencer. Combinatorica 5 241 245.