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A diffusive matrix model for invariant $β$-ensembles

We define a new diffusive matrix model converging towards the $β$-Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-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 $β< 1$ 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.