Random optimization problems at fixed temperatures

TL;DR

This paper analyzes disordered mean-field combinatorial optimization problems at fixed temperature, establishing probabilistic limit theorems for key quantities and demonstrating replica symmetry across several classical models.

Contribution

It provides the first comprehensive probabilistic limit theorems for the log-partition function and typical configurations in fixed-temperature disordered models, including applications to well-known combinatorial problems.

Findings

Law of Large Numbers and Central Limit Theorems for the log-partition function

Quenched Poisson convergence for intersection sizes

Replica symmetry demonstrated in models like TSP and minimal matching

Abstract

This article considers a class of disordered mean-field combinatorial optimization problems. We focus on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution under mild assumptions. Our results consist of the Law of Large Numbers and Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the Gibbs average in both quenched and annealed forms. We also derive quenched Poisson convergence for the size of the intersection of two independent samples, yielding replica symmetry of the model. Applications cover popular models from the literature, such as the Minimal Matching Problem, Traveling Salesman Problem, and Minimal Spanning Tree Problem, on a sequence of deterministic and random dense block graphs of increasing size.