Zero-one laws for dependent images

Bernard Coupier (1) , Paul Doukhan (2, 3, 4) , Bernard Ycart (1, 5)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 CREST - Centre de Recherche en Économie et Statistique
3 CES - Centre d'économie de la Sorbonne
4 SAMOS - Statistique Appliquée et MOdélisation Stochastique
5 LMC - IMAG - Laboratoire de Modélisation et Calcul

A set of binary random variables indexed by a lattice torus is considered. Under mixing hypothesis, the probability of any proposition belonging to the first order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to $-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given.

Mots clés: zero-one law; first-order logic; random field; weak dependance; Ising model

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Bernard Coupier, Paul Doukhan, Bernard Ycart. Zero-one laws for dependent images. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2006, 2, pp.157-175. ⟨hal-00267625⟩