Anirban Basak

Limiting spectral distributions of some band matrices

We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some further properties of the limits.

Circular law for the sum of random permutation matrices

Let

Limiting spectral distribution of sample autocovariance matrices

P_n^1,\dots, P_n^d

Diffusion limit for the partner model at the critical value

be

Limiting spectral distribution of sum of unitary and orthogonal matrices

n\times n

Invertibility of sparse non-Hermitian matrices

permutation matrices drawn independently and uniformly at random, and set

BALANCED RANDOM TOEPLITZ AND HANKEL MATRICES

S_n^d:=\sum_{\ell=1}^d P_n^\ell

The circular law for sparse non-Hermitian matrices

. We show that if

Ferromagnetic Ising measures on large locally tree-like graphs

\log^{12}n/(\log \log n)^{4} \le d=O(n)

The Evolving Voter Model on Thick Graphs

, then the empirical spectral distribution of