Convergence for noncommutative rational functions evaluated in random matrices

Regular and positive noncommutative rational functions

Call a noncommutative rational function r regular if it has no singularities, i.e., r(X) is defined for all tuples of self-adjoint matrices X. In this talk regular noncommutative rational functions r will be characterized via the properties of their (minimal size) linear systems realizations r = c∗L−1b. Our main result states that r is regular if and only if L = A0 + ∑ j Ajxj is privileged. Rou...

Rational decisions, random matrices and spin glasses

We consider the problem of rational decision making in the presence of nonlinear constraints. By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio optimisation problem. We show that the number of optimal solutions is generally exponentially large, and each of them is fragile: rationality is in this case of limited use. In addition, this problem is related ...

From random matrices to random analytic functions

Singular points of random matrix-valued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n× n matrices. Singular points are those (random) z where the matrix becomes singular, that is, the zeros of the determinant. This notion was introduced in the...

Random Polynomials, Random Matrices and L-functions

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.

Dominated Convergence for Fuzzy Random Variables