Maurice Duits

Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight

We study polynomials that are orthogonal with respect to a varying quartic weight exp(− N(x2/2 + tx4/4)) for t < 0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its and Kitaev showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann–Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Ψ-functions for the Painleve I equation in the construction of the local parametrix.

The Hermitian two matrix model with an even quartic potential

We consider the two matrix model with an even quartic potential W(y) = y^4/4 + αy^2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M_1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M_1. Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

An Equilibrium Problem for the Limiting Eigenvalue Distribution of Rational Toeplitz Matrices

We study the limiting eigenvalue distribution of

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

n\times n

Limits of determinantal processes near a tacnode

banded Toeplitz matrices as

On Mesoscopic Equilibrium for Linear Statistics in Dyson’s Brownian Motion

n\to \infty

A critical phenomenon in the two-matrix model in the quartic/quadratic case

. From classical results of Schmidt, Spitzer, and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as

On global fluctuations for non-colliding processes

n\to\infty

Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble

. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimizes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.

Powers of large random unitary matrices and Toeplitz determinants

We consider the two sequences of biorthogonal polynomials (p_(k,n))^∞_(k=0) and (q_(k,n)) ^∞_(k=0) related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_(n,n) as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.