Distribution of Eigenvalues of Ensembles of Asymmetrically Diluted Hopfield Matrices

TL;DR

This paper derives the eigenvalue distribution of large asymmetrically diluted Hopfield matrices using Grassmann variables and electrostatic analogies, revealing a uniform distribution in a circle under strong dilution.

Contribution

It introduces a novel analytical approach to compute eigenvalue distributions of diluted Hopfield matrices in the large size limit.

Findings

Eigenvalue distribution is uniform in a circle in the complex plane under strong dilution.

Method employs Grassmann variables and electrostatic analogy for analysis.

Results extend understanding of spectral properties in neural network models.

Abstract

Using Grassmann variables and an analogy with two dimensional electrostatics, we obtain the average eigenvalue distribution $\rho(\omega)$ of ensembles of $N \times N$ asymmetrically diluted Hopfield matrices in the limit $N \rightarrow \infty$. We found that in the limit of strong dilution the distribution is uniform in a circle in the complex plane.