A diffusive matrix model for invariant $\beta$-ensembles

Abstract

We define a new diffusive matrix model converging towards the $\beta$-Dyson Brownian motion for all $\beta \in [0,2]$ that provides an explicit construction of $\beta$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $\beta <1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues.

References in corpus (2)

• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

Cited by in corpus (9)

• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Eigenvector dynamics under free addition
• From Sine kernel to Poisson statistics
• The high temperature crossover for general 2D Coulomb gases
• A limit theorem at the spectral edge for corners of time-dependent Wigner matrices
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Functional central limit theorems for multivariate Bessel processes in the freezing regime
• Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \ge 1$
• On the Spontaneous Breaking of U(N) symmetry in invariant Matrix Models