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

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.


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.)
Full Text: DOI arXiv


[1] Andjel, E.D.: Characteristic exponents for two-dimensional bootstrap percolation. Ann. Probab. 21(2), 926-935 (1993) · Zbl 0787.60120
[2] Antal, P., Pisztora, A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036-1048 (1996) · Zbl 0871.60089
[3] Billingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1995) · Zbl 0822.60002
[4] Bissacot, R., Fernández, R., Procacci, A.: On the convergence of cluster expansions for polymer gases. J. Stat. Phys. 139(4), 598-617 (2010) · Zbl 1196.82135
[5] Bissacot, R., Fernández, R., Procacci, A., Scoppola, B.: An improvement of the Lovász Local Lemma via cluster expansion. Comb. Probab. Comput. (2011). doi:10.1017/S0963548311000253 · Zbl 1233.05196
[6] Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, Cambridge (2006) · Zbl 1118.60001
[7] Dobrushin, R. L., Perturbation methods of the theory of Gibbsian fields, Saint-Flour, 1994, Berlin · Zbl 0871.60086
[8] Erdős, P.; Lovász, L., Problems and results on 3-chromatic hypergraphs and some related questions, No. 10, 609-627 (1975), Amsterdam
[9] Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123-140 (2007) · Zbl 1206.82148
[10] Fisher, D.C., Solow, A.E.: Dependence polynomials. Discrete Math. 82(3), 251-258 (1990) · Zbl 0721.05036
[11] Grimmett, G.: Percolation, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999) · Zbl 0926.60004
[12] Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133-161 (1971)
[13] Hoede, C., Li, X.L.: Clique polynomials and independent set polynomials of graphs. Discrete Math. 125(1-3), 219-228 (1994). 13th British Combinatorial Conference (Guildford, 1991) · Zbl 0799.05030
[14] Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics. Springer, Berlin (2005). Reprint of the 1985 original · Zbl 1103.82016
[15] Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25(1), 71-95 (1997) · Zbl 0882.60046
[16] Mathieu, P., Temmel, C.: K-independent percolation on trees. Stoch. Process. Appl. (2012). doi:10.1016/j.spa.2011.10.014 · Zbl 1245.82026
[17] Russo, L.: An approximate zero-one law. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61(1), 129-139 (1982) · Zbl 0501.60043
[18] Scott, A.D., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118(5-6), 1151-1261 (2005) · Zbl 1107.82013
[19] Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241-245 (1985) · Zbl 0587.60012
[20] Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423-439 (1965) · Zbl 0135.18701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.