A diffusive matrix model for invariant β-ensembles

Abstract

We define a new diffusive matrix model converging towards the β -Dyson Brownian motion for all β ∈ [0 , 2] that provides an explicit construction of β -ensembles of ran- dom matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when β < 1 and that two eigenvalues collide, the eigenvectors of these two colliding eigenval- ues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues. Keywords: random matrices; stochastic calculus; Interacting particles system; Dyson Brownian motion.