On eigenvalues of the Brownian sheet matrix

Jian Song, Yimin Xiao and Wangjun Yuan

Stochastic Processes and their Applications, 2023, vol. 166, issue C

Abstract: We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence Ld(s,t),(s,t)∈[0,S]×[0,T]d∈N of empirical spectral measures of the rescaled matrices is tight on C([0,S]×[0,T],P(R)) and hence is convergent as d goes to infinity by Wigner’s semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

Keywords: Random matrix; Brownian sheet; Empirical spectral measure; High-dimensional limit; Dyson Brownian motion; McKean–Vlasov equation (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.104231

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