On the zero-temperature limit of Gibbs states

TL;DR

This paper demonstrates that for certain Lipschitz potentials on the full shift, Gibbs measures do not necessarily converge as temperature approaches zero, revealing complex zero-temperature limits in statistical mechanics models.

Contribution

The authors construct explicit examples of Lipschitz potentials where Gibbs measures fail to converge at zero temperature, including finite-range interactions in higher dimensions.

Findings

Gibbs measures can fail to converge at zero temperature for Lipschitz potentials.

Non-convergence occurs even with finite-range interactions in dimensions three and higher.

The zero-temperature limit behavior is more complex than previously understood.

Abstract

We exhibit Lipschitz (and hence H\"older) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.