Pseudo-Wigner Matrices

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 × 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.

On the spectral norms of pseudo-wigner and related matrices

“…In this section, we recall some definitions from [15] and introduce a family of pseudo-Marchenko-Pastur (pseudo-MP) ensembles analogous to the pseudo-Wigner matrices.…”

“…Next we provide an explicit constructions of the pseudo-Wigner and pseudo-MP ensembles from dual BCH codes. The idea was presented in [15] for the r-independent pseudo-Wigner matrices with r of the order of log 2 N . Here we focus on higher levels of independence with r ∝ N ρ , ρ > 0.…”

“…Given these results, the pseudo-Wigner matrices are built as explained in Section as IV of [15]. Pseudo-MP matrices are constructed analogously, by first packing the codewords of the dual BCH code row by row into rectangular N × p matrices Y N scaled by 1 √ N .…”

“…Specific pseudo-random constructions usually develop from a set of properties mimicking truly random data, and attempt to come up with deterministic ways of reproducing these properties. Following this approach, in [15] we proposed a framework allowing construction of symmetric sign matrices of low Kolmogorov complexity with spectra converging to the semicircular law. Here we extend the ideas of [15] to the construction of low complexity SCMs with spectra converging to Marchenko-Pastur law.…”

“…Following this approach, in [15] we proposed a framework allowing construction of symmetric sign matrices of low Kolmogorov complexity with spectra converging to the semicircular law. Here we extend the ideas of [15] to the construction of low complexity SCMs with spectra converging to Marchenko-Pastur law. In a related work [16], the authors show that if the columns of matrices are randomly chosen from a properly designed binary code, their spectra converge to Marchenko-Pastur law as is the case for our construction.…”