Zero-temperature phase diagram for double-well type potentials in the summable variation class

TL;DR

This paper characterizes the zero-temperature phase diagram for a class of long-range potentials with summable variation, focusing on the convergence of Gibbs measures to ground states in a binary shift space.

Contribution

It provides a complete characterization of the zero-temperature phase diagram for summable variation potentials with two ground states, using the Peierls barrier.

Findings

Identifies conditions for convergence of Gibbs measures at zero temperature.

Describes the phase diagram in terms of the Peierls barrier.

Analyzes long-range potentials on a binary shift space.

Abstract

We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols $\{0,1\}$. These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are locally constant, Lipschitz continuous or, more generally, of summable variation. We assume there exists exactly two ground states: the fixed points $0^\infty$ and $1^\infty$. We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero.