# Pseudo-Wigner Matrices

## TL;DR

This paper introduces pseudo-Wigner matrices that mimic Wigner's spectral distribution, providing explicit constructions with low complexity and demonstrating their strong pseudo-random properties through theoretical analysis and simulations.

## Contribution

The paper presents a novel explicit construction of pseudo-Wigner matrices using dual BCH codes, with provable spectral closeness to the semicircular law and low Kolmogorov complexity.

## Key findings

- Spectra of pseudo-Wigner matrices closely follow the semicircular law.
- Constructed matrices have Kolmogorov complexity of order log(N).
- Pseudo-Wigner matrices outperform quasi-random graphs in randomness tests.

## Abstract

We consider the problem of generating pseudo-random matrices based on the similarity of their spectra to Wigner's semicircular law. We introduce the notion of an r-independent pseudo-Wigner matrix ensemble and prove closeness of the spectra of its matrices to the semicircular density in the Kolmogorov distance. We give an explicit construction of a family of N by N pseudo-Wigner ensembles using dual BCH codes and show that the Kolmogorov complexity of the obtained matrices is of the order of log(N) bits for a fixed designed Kolmogorov distance precision. We compare our construction to the quasi-random graphs introduced by Chung, Graham and Wilson and demonstrate that the pseudo-Wigner matrices pass stronger randomness tests than the adjacency matrices of these graphs (lifted by the mapping 0 -> 1 and 1 -> -1) do. Finally, we provide numerical simulations verifying our theoretical results.