Constrained Eigenvalues Density of Invariant Random Matrices Ensembles

Abstract: We compute exact asymptotic of the statistical density of random matrices belonging to invariant random matrices ensemble (RMT) orthogonal, unitary and symplectic ensembles, where all its eigenvalues lie within the interval $[\sigma, +\infty[$ or $]-\infty,\tau]$ or $[\sigma,\tau]$. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied by Dean-Majumdar.