Positivity of Gibbs states on distance-regular graphs

2021/11/03 by Michael Voit, Voit, Michael
Computer Science · Mathematics · #05E30 #20N20 #33C45 #43A62 #43A90 #81R12 #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2111.02116

openalex publication_date 2021/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈[-1,1] the function n↦ xⁿ 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.

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