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A random walk approach to linear statistics in random tournament ensembles. (English) Zbl 1398.05094
Summary: 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.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C80 Random graphs (graph-theoretic aspects) 05C81 Random walks on graphs 15B52 Random matrices (algebraic aspects)
##### Keywords:
random matrix theory; random walks
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