Annealed averages in spin and matrix models

TL;DR

This paper explores annealed averages in disordered systems, particularly in matrix models, highlighting their role in developing planted solutions and analyzing matrix element distributions, contrasting with the more common quenched approach.

Contribution

It introduces the study of annealed averages in matrix models and demonstrates their connection to planted solutions and inference problems, expanding understanding beyond traditional quenched systems.

Findings

Annealed systems develop spontaneous planted solutions.

Distribution of matrix elements can be analyzed via annealed averages.

Connections to inference problems are established.

Abstract

A disordered system is denominated `annealed' when the interactions themselves may evolve and adjust their values to lower the free energy. The opposite (`quenched') situation when disorder is fixed, is the one relevant for physical spin-glasses, and has received vastly more attention. Other problems however are more natural in the annealed situation: in this work we discuss examples where annealed averages are interesting, in the context of matrix models. We first discuss how in practice, when system and disorder adapt together, annealed systems develop `planted' solutions spontaneously, as the ones found in the study of inference problems. In the second part, we study the probability distribution of elements of a matrix derived from a rotationally invariant (not necessarily Gaussian) ensemble, a problem that maps into the annealed average of a spin glass model.