Random optimization problems at fixed temperatures

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 graphs of increasing size.