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Maurice Duits

Publications and source records attributed to Maurice Duits.

32 records · Page 2Linked to original sources
arXiv2013

Painlevé kernels in Hermitian matrix models

After reviewing the Hermitian one matrix model, we will give a brief introduction to the Hermitian two matrix model and present a summary of some recent results on the asymptotic behavior of the two matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a $4\times 4$ Riemann-Hilbert problem related to Painlevé II equation. Also an open problem will be presented.

math-ph↗
arXiv2013

The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles

We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers (analogous to the recently proven local semicircle law for Wigner matrices) under fairly weak conditions on the underlying measure $μ$. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.

math.PR↗
arXiv2012

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

We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes with an exponent 1/2 in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4 x 4 Riemann-Hilbert problem related to the Hastings-McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in non-colliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes with an exponent 1/2 at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.

math-ph↗
arXiv2012

Nonintersecting paths with a staircase initial condition

We consider an ensemble of $N$ discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as $N\to \infty$. In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.

math.PR↗
arXiv2011

The Gaussian free field in an interlacing particle system with two jump rates

We study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field.

math-ph↗
arXiv2010

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+alpha y^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 alpha=0, where the external field on the third measure was not present.

math-ph↗
arXiv2010

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

We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k 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 tends to infinity. 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 behavior for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

math.CA↗
arXiv2009

An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize the recent results of Duits and Kuijlaars for banded Toeplitz matrices.

math.CV↗
arXiv2009

Limits of determinantal processes near a tacnode

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter $\eps> 0$. The domain has two cusps, one pointing up and one pointing down. In the limit $\varepsilon\downarrow 0$ the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime $\varepsilon \downarrow 0$. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

math.PR↗
arXiv2008

Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis

The eigenvalue statistics of a pair $(M_1,M_2)$ of $n\times n$ Hermitian matrices taken random with respect to the measure $$\frac{1}{Z_n}\exp\big(-n\Tr (V(M_1)+W(M_2)-τM_1M_2)\big) {\rm d}M_1 {\rm d} M_2 $$ can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest descent analysis of a $4 \times 4$ matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case $W(y)=y^4/4$ and $V$ an even polynomial. As a result we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of $M_1$ (when averaged over $M_2$) in the global and local regime as $n\to \infty$ in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

math-ph↗
arXiv2007

Powers of large random unitary matrices and Toeplitz determinants

We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $ \Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.

math-ph↗
arXiv2007

An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices

We study the limiting eigenvalue distribution of $n\times n$ banded Toeplitz matrices as $n\to \infty$. 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 $n\to\infty$. 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 minimes 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.

math.CV↗
arXiv2006

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(x^2/2+tx^4/4)) 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 Psi-functions for the Painleve I equation in the construction of the local parametrix.

math.CA↗