I will discuss sums of Hermitian random matrices $M + \sqrt{\tau} H$
where $M$ is a matrix from a unitary ensemble of the form
$$ \frac{1}{Z_n} e^{-n Tr V(M)}$$ and $H$ is a scaled GUE matrix.
For very special choices of the potential $V$ the eigenvalues of $M$ has
a density that vanishes at an interior point, or vanishes to higher
order at an edge point. We show that the singular behavior persists for the eigenvalues of $M + \sqrt{\tau} H$ for $\tau$ up to a critical value $\tau_{cr}$.
In addition, the local scaling limits at the singular point continue to hold as
well. This is joint work with Tom Claeys, Karl Liechty and Dong Wang

In collaboration with Francesco Mezzadri and Fabio Deelan Cunden

[J. Phys. A: Math. Theor. 48, 315204 (2015); Phys. Rev. Lett. 113, 070202 (2014)]

This talk concerns with the limiting behaviour of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or the arcsine distribution, respectively. Regarding that
convergence as the law of large number, a central limit theorem is then derived.

The probability distribution of spectral moments for the Gaussian beta-ensembles