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arXiv · 1711.02072

A random walk approach to linear statistics in random tournament ensembles

Christopher H. Joyner·Uzy Smilansky
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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.

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Christopher H. Joyner, Uzy Smilansky. 2017-11-06. A random walk approach to linear statistics in random tournament ensembles. https://arxiv.org/abs/1711.02072

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