A random walk approach to linear statistics in random tournament ensembles

Abstract

We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq}={\overline{H}}_{qp}=\pm i$, that are either independently distributed or exhibit global correlations imposed by the condition ${\sum }_q{H}_{pq}=0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein's method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.

33 pages, 4 figures