Stochastic symmetry-breaking in a gaussian Hopfield model

TL;DR

This paper investigates a Gaussian Hopfield model with two patterns, revealing an infinite number of pure states at low temperatures due to stochastic symmetry-breaking caused by disorder.

Contribution

It demonstrates the existence of infinitely many pure states and the support of the metastate on symmetric pairs, highlighting a novel symmetry-breaking mechanism.

Findings

Infinite pure states at low temperatures

Metastate supported on symmetric pairs

Random symmetry-breaking of disorder distribution

Abstract

We study a ``two-pattern'' Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and we find that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder