Positivity of Gibbs states on distance-regular graphs

TL;DR

This paper establishes criteria for the positivity of Gibbs states on distance-regular graphs, utilizing polynomial hypergroup theory and embedding into graph families, with applications to Hamming and infinite distance-transitive graphs.

Contribution

It introduces a new criterion for Gibbs state positivity on distance-regular graphs based on graph embeddings and polynomial hypergroup theory, providing complete descriptions for specific graph classes.

Findings

Complete characterization of positive Gibbs states on Hamming graphs

Complete description of positive Gibbs states on infinite distance-transitive graphs

Development of a criterion linking positivity to orthogonal polynomial representations

Abstract

We study criteria which ensure that Gibbs states (often also called generalized vacuum states) on distance-regular graphs are positive. Our main criterion assumes that the graph can be embedded into a growing family of distance-regular graphs. For the proof of the positivity we then use polynomial hypergroup theory and translate this positivity into the problem whether for $x\in[-1,1]$ the function $n\mapsto x^n$ has a positive integral representation w.r.t. the orthogonal polynomials associated with the graph. We apply our criteria to several examples. For Hamming graphs and the infinite distance-transitive graphs we obtain a complete description of the positive Gibbs states.