Product type potential on the $X\,Y$ model: selection of maximizing probability and a large deviation principle

TL;DR

This paper analyzes the $XY$ model with product type potentials, deriving explicit equilibrium probabilities, exploring ergodic optimization and maximizing probabilities, and establishing a large deviation principle as temperature approaches zero.

Contribution

It provides explicit formulas for equilibrium states and deviation functions in the $XY$ model with product potentials, advancing understanding of zero-temperature limits.

Findings

Explicit equilibrium probability expressions

Selection of maximizing probability as temperature approaches zero

Large deviation principle with an explicit deviation function

Abstract

Given an interval $[a,b]$ the associated $X\,Y$ model is the space $\Omega=[a,b]^\mathbb{N}$ with an a priori probability $\nu $ on the state space $[a,b]$. We will present here the case of the product type potential on the $X\,Y$ model and in this setting we can show the explicit expression of the equilibrium probability. We will also consider questions about Ergodic Optimization, maximizing probabilities, subactions and we will show selection of a maximizing probability, when temperature goes to zero. Finally we show a large deviation principle when temperature goes to zero and we present an explicit expression for the deviation function.