Generalization of the cavity method for adiabatic evolution of Gibbs states

TL;DR

This paper extends the cavity method to analyze how Gibbs states in mean field glassy systems evolve as external parameters like temperature change, providing insights into slow annealing, temperature chaos, and optimization transitions.

Contribution

It introduces a generalized cavity method for adiabatic evolution of Gibbs states, with detailed solutions for specific models and connections to other theoretical frameworks.

Findings

Demonstrates temperature chaos in glassy systems

Identifies easy/hard transition in simulated annealing

Analyzes slow Monte-Carlo annealing behavior

Abstract

Mean field glassy systems have a complicated energy landscape and an enormous number of different Gibbs states. In this paper, we introduce a generalization of the cavity method in order to describe the adiabatic evolution of these glassy Gibbs states as an external parameter, such as the temperature, is tuned. We give a general derivation of the method and describe in details the solution of the resulting equations for the fully connected p-spin model, the XOR-SAT problem and the anti-ferromagnetic Potts glass (or "coloring" problem). As direct results of the states following method, we present a study of very slow Monte-Carlo annealings, the demonstration of the presence of temperature chaos in these systems, and the identification of a easy/hard transition for simulated annealing in constraint optimization problems. We also discuss the relation between our approach and the Franz-Parisi potential, as well as with the reconstruction problem on trees in computer science. A mapping between the states following method and the physics on the Nishimori line is also presented.