Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models

TL;DR

This paper characterizes the sets of ground states in uniformly chaotic finite-range lattice models, showing they are topologically closed, connected, and belong to the $\

Contribution

It provides a complete topological and computational classification of ground states in uniformly chaotic models, linking them to the arithmetical hierarchy.

Findings

Ground states are topologically closed and connected.

Sets of ground states are $\

Every $\

Abstract

Chaotic dependence on temperature refers to the phenomenon of divergence of Gibbs measures as the temperature approaches a certain value. Models with chaotic behaviour near zero temperature have multiple ground states, none of which are stable. We study the class of uniformly chaotic models, that is, those in which, as the temperature goes to zero, every choice of Gibbs measures accumulates on the entire set of ground states. We characterise the possible sets of ground states of uniformly chaotic finite-range models up to computable homeomorphisms. Namely, we show that the set of ground states of every model with finite-range and rational-valued interactions is topologically closed and connected, and belongs to the class $\Pi_2$ of the arithmetical hierarchy. Conversely, every $\Pi_2$-computable, topologically closed and connected set of probability measures can be encoded (via a computable homeomorphism) as the set of ground states of a uniformly chaotic two-dimensional model with finite-range rational-valued interactions.