# Description of some ground states by Puiseux technics

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1 IMECC - Instituto de Matemática, Estatística e Computação Científica Brésil 2 IMB - Institut de Mathématiques de Bordeaux

Abstract : Let $\Sigma^+ G, \sigma$ be a one-sided transitive subshift of finite type, where symbols are given by a finite spin set $ S $, and admissible transitions are represented by an irreducible directed graph $ G\subset S\times S $. Let $ H : \Sigma^+ G\to\mathbb{R}$ be a locally constant function that corresponds with a local observable which makes finite-range interactions. Given $\beta > 0$, let $ \mu {\beta H} $ be the Gibbs-equilibrium probability measure associated with the observable $-\beta H$. It is known, by using abstract considerations, that $\{\mu {\beta H}\} {\beta>0}$ converges as $ \beta \to + \infty $ to a $H$-minimizing probability measure $\mu {\textrm{min}}^H$ called zero-temperature Gibbs measure. For weighted graphs with a small number of vertices, we describe here an algorithm similar to the Puiseux algorithm that gives the explicit form of $\mu {\textrm{min}}^H$ on the set of ground-state configurations

Keywords : zero-temperature Gibbs measures ground-state configurations Puiseux algorithm

Author: ** Eduardo Garibaldi - Philippe Thieullen - **

Source: https://hal.archives-ouvertes.fr/