# Annealed Ising model on configuration models

## TL;DR

This paper investigates the annealed Ising model on configuration graphs, revealing how degree structure influences the critical temperature and providing a rigorous variational formula for the pressure, contrasting some physics conjectures.

## Contribution

It offers the first rigorous analysis of the annealed Ising model on configuration models, highlighting the impact of degree distribution on critical behavior.

## Key findings

- Deterministic degrees yield the same critical value as quenched models.
- i.i.d. degrees lead to a smaller annealed critical value.
- Provides a variational formula for the annealed pressure.

## Abstract

In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we take the average on both sides of the ratio {defining the Boltzmann-Gibbs measure of the Ising model}. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.