Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

Abstract

AbstractLet A be a finite set and let ϕ:Aℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μβϕ. We prove that as β→+∞, the family (μβϕ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.