One object of interest in random matrix theory is a family of point ensembles (random point configurations) related to various systems of classical orthogonal polynomials. The paper deals with a one--parametric deformation of these ensembles, which is defined in terms of the biorthogonal polynomials of Jacobi, Laguerre and Hermite type.
Our main result is a series of explicit expressions for the correlation functions in the scaling limit (as the number of points goes to infinity). As in the classical case, the correlation functions have determinantal form. They are given by certain new kernels which are described in terms of the Wright's generalized Bessel function and can be viewed as a generalization of the well--known sine and Bessel kernels.
In contrast to the conventional kernels, the new kernels are non--symmetric. However, they possess other, rather surprising, symmetry properties.
Our approach to finding the limit kernel also differs from the conventional one, because of lack of a simple explicit Christoffel--Darboux formula for the biorthogonal polynomials.
The Muttalib-Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $θ$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $θ$ there is an equivalent vector equilibrium problem.
The Muttalib-Borodin biorthogonal ensemble is a probability density function for $n$ particles on the positive real line that depends on a parameter $θ$ and an external field $V$. For $θ=\frac{1}{2}$ we find the large $n$ behavior of the associated correlation kernel with only few restrictions on $V$. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a $3\times 3$ Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the orig…
We investigate determinantal point processes on $[0,+\infty)$ of the form \begin{equation*}\label{probability distribution} \frac{1}{Z_n}\prod_{1\leq i<j\leq n}(λ_j-λ_i)\prod_{1\leq i<j\leq n}(λ_j^θ-λ_i^θ) \prod_{j=1}^n w(λ_j)dλ_j,\qquad θ\geq 1. \end{equation*} We prove that the biorthogonal polynomials associated to such models satisfy a recurrence relation and a Christoffel-Darboux formula if $θ\in\mathbb Q$, and that they can be characterized in terms of $1\times 2$ vector-valued Riemann-Hilbert problems which exhibit some non-standard properties. In addition, we obtain expressions for the…
We consider sequences of biorthogonal polynomials with respect to a Cauchy type convolution kernel and give the weak and ratio asymptotic of the corresponding sequences of biorthogonal polynomials. The construction is intimately related with a mixed type Hermite-Padé approximation problem whose asymptotic properties is also revealed.
This paper is a continuation of the recent paper "CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus transformations" by the same authors. The behavior of quasidefinite sesquilinear forms for Laurent polynomials in the complex plane, characterized by bivariate linear functionals, and corresponding CMV biorthogonal Laurent polynomial families --including Sobolev and discrete Sobolev orthogonalities--- under two type of Geronimus--Uvarov transformations is studied. Either the linear functionals are multiplied by a Laurent polynomial and divided by the comple…
Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities like Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a div…
A finite family of $R_I$ polynomials is introduced and studied. It consists in a set of polynomials of $_{3}F_{2}$ form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of $R_I$ type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions.
We consider $n$ particles $0\leq x_1<x_2< \cdots < x_n < +\infty$, distributed according to a probability measure of the form $$ \frac{1}{Z_n}\prod_{1\leq i <j \leq n}(x_j-x_i)\prod_{1\leq i <j \leq n}(x_j^θ-x_i^θ)\prod_{j=1}^nx_j^αe^{-x_j}\ud x_j, ~~ α>-1,~~ θ>0, $$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which all…
The Gauss-Borel or $LU$ factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the spectral matrices modeling the multiplication by the independent variable $x$, the Christoffel-Darboux kernel and its projection properties, are discussed from this point of view. Then, the Hankel case is presented and different properties, specific of this case, as the three terms relations, Heine formulas, Gauss quadrature and the Christoffel-D…
In this paper, we consider Muttalib-Borodin ensemble of Laguerre type, a determinantal point process over $[0,\infty)$ which depends on the varying weights $x^αe^{-nV(x)}$, $α>-1$, and a parameter $θ$. For $θ$ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions $V$ as $n\to \infty$. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in the special cases and conjectured to be universal. Our proof is based on the Deift/Zhou nonline…
One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
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 calculat…
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) …
Rosengren and Schlosser introduced notions of ${\it R}_N$-theta functions for the seven types of irreducible reduced affine root systems, ${\it R}_N={\it A}_{N-1}$, ${\it B}_{N}$, ${\it B}^{\vee}_N$, ${\it C}_{N}$, ${\it C}^{\vee}_N$, ${\it BC}_{N}$, ${\it D}_{N}$, $N \in \mathbb{N}$, and gave the Macdonald denominator formulas. We prove that, if the variables of the ${\it R}_N$-theta functions are properly scaled with $N$, they construct seven sets of biorthogonal functions, each of which has a continuous parameter $t \in (0, t_{\ast})$ with given $0< t_{\ast} < \infty$. Following the standar…
We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakl…
In this paper, we study the singularly perturbed Laguerre unitary ensemble $$ \frac{1}{Z_n} (\det M)^αe^{- \textrm{tr}\, V_t(M)}dM, \qquad α>0, $$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of $t/x$ for varying $t$, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $ψ$-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the …
We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a,0] and [0,1], for a < 0. As a \to -1 the particles around 0 experience a phase transition. This transition is studied in a double scaling limit, where we let the number of particles of the ensemble tend to infinity while the parameter a tends to -1 at a rate of order n^{-1/2}. The correlation kernel converges, in this regime, to a new kind of universal kernel, the Angelesco kernel K^{Ang}. The result follows from the Deift/Zhou steepest descent analysis, applied to the Riemann-Hilbert problem f…
We survey the current status of universality limits for $m$-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider $β$ ensembles for $β\neq 2$, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.
Let $X$ 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 $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ 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 $H + M$ where $H$ is a GUE …
We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on the sphere. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated to isotropic …
The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $τ$-function, in the sense of Okamoto, for the Painlevé system \PIII. This then leads to a factorisation of the probability as the product of two $τ$-functions for the Painlevé system \PIIIdash. A previous study has given a formula of thi…
We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β= 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N \to \infty, of the associated kernel in the neighborhood of a point on the boundary under the assum…
The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^α(1+x)^βh(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the…
The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the Christoffel-Darboux kernel [59] become the key for studying universality. In this paper the existing results on the CD-kernel will be extended in two directions. Firstly, in order to analyze the transition from the universal to the non-universal regime, we provide leading order asymptotics that are global rather than local. This allows e.g. to describe the …
