A random walk approach to linear statistics in random tournament ensembles

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.