Distribution of Eigenvalues of Ensembles of Asymmetrically Diluted Hopfield Matrices

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.