Polynomial Ensembles Research Articles

Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures

This paper is the first in a series of three. We give an explicit description of the ergodic decomposition of infinite Pickrell measures on the space of infinite complex matrices. A key role is played by the construction of -finite analogues of determinantal measures on spaces of configurations, including the infinite Bessel process, a scaling limit of the -finite analogues of the Jacobi orthogonal polynomial ensembles. Our main result identifies the infinite Bessel process with the decomposing measure of an infinite Pickrell measure.

Correlation kernels for sums and products of random matrices

Let [Formula: see text] be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of [Formula: see text] and [Formula: see text], where [Formula: see text] is a complex Ginibre matrix and [Formula: see text] is a truncated unitary matrix. We also consider the product of [Formula: see text] and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum [Formula: see text] where [Formula: see text] is a GUE matrix and [Formula: see text] is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of [Formula: see text] follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.

Dynamical correlation functions for products of random matrices

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.

On the number of connected components of random algebraic hypersurfaces

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d \rightarrow \infty$), we prove that the order of growth of $\mathbb{E} b_0(X)$ satisfies: $$\mathbb{E} b_{0}(X)=\Theta\left(\left[ \mathbb{E} b_0(X\cap \mathbb{R}P^1) \right]^{n} \right). $$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f|_{\mathbb{R}P^1}.$ The proof requires an upper bound for $\mathbb{E} b_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E} b_0(X)$ (as $n \rightarrow \infty$).

On Certain Wronskians of Multiple Orthogonal Polynomials

We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szeg\H{o}. There are two applications of our results. The first application arises from the observation that the $m$-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even $m$ in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Tur\'an type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.

Zeros of polynomials with random coefficients

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.

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 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.

Singular values of products of random matrices and polynomial ensembles

Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy–Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

A note on the limiting mean distribution of singular values for products of two Wishart random matrices

The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg, and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M = 2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.

Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles

The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group $S_\infty$. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz matrices of infinite size. The purpose of the paper is to construct a family $\{X^{(z,z')}\}$ of continuous time Markov processes on the Thoma cone, depending on two continuous parameters $z$ and $z'$. Our construction largely exploits specific properties of the Thoma cone related to its representation-theoretic origin, although we do not use representations directly. On the other hand, we were inspired by analogies with random matrix theory coming from models of Markov dynamics related to orthogonal polynomial ensembles. We show that processes $X^{(z,z')}$ possess a number of nice properties, namely: (1) every $X^{(z,z')}$ is a Feller process; (2) the infinitesimal generator of $X^{(z,z')}$, its spectrum, and the eigenfunctions admit an explicit description; (3) in the equilibrium regime, the finite-dimensional distributions of $X^{(z,z')}$ can be interpreted as (the laws of) infinite-particle systems with determinantal correlations; (4) the corresponding time-dependent correlation kernel admits an explicit expression, and its structure is similar to that of time-dependent correlation kernels appearing in random matrix theory.

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

A universality theorem for ratios of random characteristic polynomials

We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a universality limit for these ratios.

Large deviations for disordered bosons and multiple orthogonal polynomial ensembles

We prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal Laguerre, Jacobi, and Hermite ensembles, the matrix model of Lueck, Sommers, and Zirnbauer for disordered bosons, the Stieltjes-Wigert matrix model of Chern-Simons theory, and Angelesco ensembles.

Crossover ensembles of random matrices and skew-orthogonal polynomials

In a recent paper (S. Kumar, A. Pandey, Phys. Rev. E, 79, 2009, p. 026211) we considered Jacobi family (including Laguerre and Gaussian cases) of random matrix ensembles and reported exact solutions of crossover problems involving time-reversal symmetry breaking. In the present paper we give details of the work. We start with Dyson’s Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions. For arbitrary dimensions we derive the joint probability density (jpd) of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles. We focus on the orthogonal-unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew-orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter. Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed.

Average characteristic polynomials for multiple orthogonal polynomial ensembles

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann–Hilbert matrix Y consisting of four blocks Y 1 , 1 , Y 1 , 2 , Y 2 , 1 and Y 2 , 2 . In this paper, we show that det Y 1 , 1 ( det Y 2 , 2 ) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize the classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel–Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

Nonintersecting paths and the Hahn orthogonal polynomial ensemble

We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation-invariant extension of the discrete sine process, which can also be described by an ergodic Gibbs measure with appropriate parameters.

Random polynomials and (pluri)potential theory

For certain ensembles of random polynomials we give the expected value of the zero distribution (in one variable) and the expected value of the distribution of common zeros of $m$ polynomials (in $m$ variables).