Random Matrices: Theory and Applications

We extend to a situation involving matrix-valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathe-matics and signal processing. We consider a case of matrix-valued Jacobi polynomials which arises from the study of representations of $\mathrm{\text{SU}}(N)$, a group that plays an important role in Random Matrix Theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix-valued situation.

In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight

From the ladder operator approach, and the associated compatibility conditions, the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter $s$ in the weight. Then we work out the large degree asymptotics of their recurrence coefficients. We also discuss the associated Hankel determinant. We show that the logarithmic derivative of $D_n(s)$ can be expressed in terms of the recurrence coefficients, and obtained the large degree asymptotics of $ln[D_n(1)]$. Based on this result, we compute the probability that an $n\times n$ (or $rN\times rN$) random matrix from a generalized Gaussian Unitary Ensemble (gGUE) is positive definite.

This paper is a survey of results that show how, in some cases, to compute the constant term which occurs in the Strong Szegö–Widom Limit Theorem for block Toeplitz matrices. While this constant in the scalar case has a long history, it is really only in a few instances that it can be calculated explicitly in the case of matrix-valued symbols.

We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.