Open Access
Translator Disclaimer
January 1997 Domination by product measures
T. M. Liggett, R. H. Schonmann, A. M. Stacey
Ann. Probab. 25(1): 71-95 (January 1997). DOI: 10.1214/aop/1024404279


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.


Download Citation

T. M. Liggett. R. H. Schonmann. A. M. Stacey. "Domination by product measures." Ann. Probab. 25 (1) 71 - 95, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0882.60046
MathSciNet: MR1428500
Digital Object Identifier: 10.1214/aop/1024404279

Primary: 60G10 , 60G60
Secondary: 60K35

Keywords: critical points , product measures , Random fields , rescaling , stochastic domination

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
Back to Top