On eigenvalues of the Brownian sheet matrix

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.