Electronic Journal of Probability

A random walk approach to linear statistics in random tournament ensembles

Christopher H. Joyner and Uzy Smilansky

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 80, 37 pp.

Received: 6 February 2018
Accepted: 15 July 2018
First available in Project Euclid: 12 September 2018

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Zentralblatt MATH identifier

Primary: 05C20: Directed graphs (digraphs), tournaments 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs 15B52: Random matrices

random matrix theory random walks graph theory

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Joyner, Christopher H.; Smilansky, Uzy. A random walk approach to linear statistics in random tournament ensembles. Electron. J. Probab. 23 (2018), paper no. 80, 37 pp. doi:10.1214/18-EJP199. https://projecteuclid.org/euclid.ejp/1536717739

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