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 $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\mathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\mathbb R))$ 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.