Shearer’s measure and stochastic domination of product measures. (English) Zbl 1320.60067

Temmel, Christoph

J. Theor. Probab. 27, No. 1, 22-40 (2014).

Summary: Let \(G=(V,E)\) be a locally finite graph. Let \(\vec{p}\in[0,1]^{V}\). We show that J. B. Shearer’s measure [Combinatorica 5, 241–245 (1985; Zbl 0587.60012)], introduced in the context of the Lovász Local Lemma [P. Erdős and L. Lovász, in: Infinite and finite sets. Colloq. honour Paul Erdős, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 609–627 (1975; Zbl 0315.05117)], with marginal distribution determined by \(\vec{p}\), exists on \(G\) if and only if every Bernoulli random field with the same marginals and dependency graph \(G\) dominates stochastically a non-trivial Bernoulli product field. Additionally, we derive a non-trivial uniform lower bound for the parameter vector of the dominated Bernoulli product field. This generalises previous results by T. M. Liggett et al. [Ann. Probab. 25, No. 1, 71–95 (1997; Zbl 0882.60046)] in the homogeneous case, in particular on the \(k\)-fuzz of \(\mathbb Z\). Using the connection between Shearer’s measure and a hardcore lattice gas established by A. D. Scott and A. D. Sokal [J. Stat. Phys. 118, No. 5–6, 1151–1261 (2005; Zbl 1107.82013)], we transfer bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.

MSC:

60E15	Inequalities; stochastic orderings
60G60	Random fields
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05D40	Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)

Keywords:

stochastic domination; Lovász local lemma; product measure; Bernoulli random field; stochastic order; hardcore lattice gas

Citations:

Zbl 0587.60012; Zbl 0315.05117; Zbl 0882.60046; Zbl 1107.82013

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