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Potts model on infinite graphs and the limit of chromatic polynomials

Aldo Procacci·Benedetto Scoppola·Victor Gerasimov
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Abstract

Given an infinite graph $\GI$ quasi-transitive and amenable with maximum degree $\D$, we show that reduced ground state degeneracy per site $W_r(\GI,q)$ of the q-state antiferromagnetic Potts model at zero temperature on $\GI$ is analytic in the variable 1/q, whenever $|2\D e^3/q|< 1$. This result proves, in an even stronger formulation, a conjecture originally sketched in [KE] (Kim and Enting, 1979) and explicitly formulated in [ST] (Shrock and Tsai,1997), based on which a sufficient condition for $W_r(\GI,q)$ to be analytic at 1/q=0 is that $\GI$ is a regular lattice.

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Aldo Procacci, Benedetto Scoppola, Victor Gerasimov. 2002-01-11. Potts model on infinite graphs and the limit of chromatic polynomials. https://doi.org/10.1007/s00220-002-0789-x

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