Séminaire de Probabilités commun ICJ/UMPA

Uniformly positive correlations in the dimer model and phase transition in lattice permutations in Z^d, d > 2, via reflection positivity

par Lorenzo Taggi

Europe/Paris
Fokko du Cloux (La Doua, Bâtiment Braconnier)

Fokko du Cloux

La Doua, Bâtiment Braconnier

Description
This talk considers two models related to each other, the dimer model and lattice permutations. The dimer model is a statistical mechanics model whose configurations are perfect matchings of a graph, namely subsets of edges which cover every vertex precisely once; lattice permutations can be viewed as a system of directed self-avoiding loops interacting by mutual exclusion and are related to the loop O(N) model and to the quantum Bose gas. Our first main result is that correlations between monomers in the dimer model in Z^d do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is ‘long’, namely that the distance between its end-points is of the same order of magnitude as the diameter of the box. These results follow from the property of reflection positivity and from the derivation of an Infrared bound from a new framework which is related to the random current representation of the Ising model.