Some patterned matrices with independent entries

TL;DR

This paper extends the understanding of the spectral distribution of patterned random matrices by relaxing the i.i.d. assumption, revealing more general limit behaviors linked to broader classes of partitions.

Contribution

It generalizes previous results on spectral distributions of patterned matrices by relaxing the i.i.d. assumption and introducing broader classes of partitions, showing non-universality.

Findings

Limits are defined via a larger class of partitions.

Results include band, sparse, continuous, and discrete variance profile matrices.

Spectral distributions are not universal under the new assumptions.

Abstract

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread -- the $2k$th moment of the limit equals a weighted sum over different types of pair-partitions of the set $\{1, 2, \ldots, 2k\}$ and are universal. Some results are also known for the sparse case. In this paper we generalise these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.