Constrained Eigenvalues Density of Invariant Random Matrices Ensembles

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 $[\sigma ,+\mathrm{\infty }[$ or $]-\mathrm{\infty },\tau ]$ 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.