Near optimal configurations in mean field disordered systems

TL;DR

This paper introduces a novel technique based on a generalized cavity method to analyze how the energy of configurations varies with their overlap to the ground state in mean field disordered systems, applicable to problems like the random matching problem and diluted spin glasses.

Contribution

It develops a new approach extending the cavity method to systems interacting with their ground state, enabling detailed energy landscape analysis in disordered systems.

Findings

Computed the energy variation as a function of overlap with the ground state.

Analyzed the random matching problem and diluted spin glass models.

Calculated the de Almeida-Thouless transition line for spin glasses on fixed connectivity graphs.

Abstract

We present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems. Our approach is based on a generalization of the cavity method to a system interacting with its ground state. With this technique we study the random matching problem as well as the mean field diluted spin glass. As a byproduct of this approach we calculate the de Almeida-Thouless transition line of the spin glass on a fixed connectivity random graph.