Non-backtracking loop soups and statistical mechanics on spin networks

Abstract

We introduce and study a Markov field on the edges of a graph in dimension $d\geq2$ whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set, the distributions of the loops and arcs on either side of the boundary are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension.

27 pages, 1 figure. V2: New results on critical behavior added, some typos corrected, other minor changes. V3: New results added (Section 4), new presentation of main results, slight generalization of some old results, other minor changes