Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble

Abstract

We consider the symmetric tridiagonal matrix-valued process associated with Gaussian beta ensemble (G$\beta$E) by putting independent Brownian motions and Bessel processes on the diagonal entries and upper (lower)-diagonal ones, respectively. Then, we derive the stochastic differential equations that the eigenvalue processes satisfy, and we show that eigenvalues of their (indexed) principal minor sub-matrices appear in the stochastic differential equations. By the Cauchy's interlacing argument for eigenvalues, we can characterize the sufficient condition that the eigenvalue processes never collide with each other almost surely, by the dimensions of the Bessel processes.

References in corpus (5)

• Matrix Models for Beta Ensembles
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Non-Markov property of certain eigenvalue processes analogous to Dyson's model
• Noncolliding processes, matrix-valued processes and determinantal processes
• Eigenvalue processes of Elliptic Ginibre Ensemble and their Overlaps