Topics In Random Matrices

I will discuss three problems on Hermitian Random Matrices, which ultimately are about orthogonal polynomials:
1. The singularly deformed Jacobi ensembles, where an infinitely fast zero is introduced at an end point of the support of Jacobi weight, and the Hankel determinant under double scaling.
2. On the generating functions of linear statistics of the orthogonal and symplectic ensembles with Gaussian and Gamma “background” distributions.
3. The least eigenvalue of family of Hankel matrices obtained from the large $n$ asymptotic of polynomials orthogonal with respect to $\exp(-x^{\beta})$, $x\geq0$, $\beta>0$. In general, the smallest eigenvalue goes to zero rapidly, for $\beta>1/2$ and at $\beta=1/2$ it is conjectured that the smallest eigenvalue decays slowly. Comparison with numerical computation is made.
These are joint work with Min Chao, Chen Min (University of Macau), Nigel Lawrence (Imperial College), Niall Emmart and Charles C. Weems (University of Massachusetts Amherst).