Non-Backtracking Loop Soups and Statistical Mechanics on Spin Networks

Abstract

We introduce and study a Markov field on the edges of a graph $\mathcal{G}$ in dimension $\textit{d}$ ≥ 2 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 that splits the graph into two parts, the distributions of the loops and arcs contained in the two parts 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.