Non-backtracking loop soups and statistical mechanics on spin networks

TL;DR

This paper introduces a Markov field model for spin networks derived from non-backtracking loop soups, allowing exact computation of free energy and analysis of critical behavior in various dimensions.

Contribution

It presents a novel Markov field on graph edges based on non-backtracking loop soups with a spatial Markov property and exact solvability.

Findings

Exact formulas for free energy density.

Analysis of critical behavior in multiple dimensions.

Independence properties of loops and arcs across boundaries.

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.