A random walk approach to linear statistics in random tournament ensembles

2017/11/06 by Joyner, Christopher H., Smilansky, Uzy
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1711.02072

Abstract

We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form H_pq = H_qp = ± i, that are either independently distributed or exhibit global correlations imposed by the condition ∑_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.

Related

• Spectral statistics of Bernoulli matrix ensembles - a random walk approach (I)
• Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs
• Tournaments and random walks
• Universal Gaussian fluctuations of non-Hermitian matrix ensembles
• Theory of random matrices with strong level confinement