We outline a relation between the densities for the $β$-ensembles with respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval $(-1,1)$ and the Cauchy weight $(1-\mathrm{i}x)^η(1+\mathrm{i}x)^{\barη}$ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for $β=2$, and of degree five for $β=1$ and $4$, analogues of which are already known for the Jacobi weight $x^a(1-x)^b$ supported on $(0,1)$. We concentrate on the case $a=b$ (Jacobi weight on $(-1,1)$) and $η$ real (Cauchy weight) since the density is then an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on $(-1,1)$ and/or the moments of the Cauchy weight. Particular attention is paid to the case $β=2$ and the Jacobi weight on $(-1,1)$ in the symmetric case $a=b$, which in keeping with a recent result obtained by Assiotis et al.~for the $β=2$ case of the symmetric Cauchy weight (parameter $η$ real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with $η=-β(N-1)/2-1-α$, after appropriately scaling $α$ proportional to $N$, we use differential equations to compute terms in the $1/N^2$ ($1/N$) expansion of the resolvent for $β=2$ ($β=1,4$).
Recently, M. de León el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
We calculate the $k$-point generating function of the correlated Jacobi ensemble using supersymmetric methods. We use the result for complex matrices for $k=1$ to derive a closed-form expression for eigenvalue density. For real matrices we obtain the density in terms of a twofold integral that we evaluate numerically. For both expressions we find agreement when comparing with Monte Carlo simulations. Relations between these quantities for the Jacobi and the Cauchy-Lorentz ensemble are derived.
A $θ$-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib-Borodin ensemble. The $θ$-deformed Cauchy-Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written as the Fox H-functions, generalising the Meijer G-function class known from the study of the case $θ= 1$. In the $θ=1$ case, it is shown Laguerre-Bures ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding the Bures e…
In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^α(1-x)^β,~x\in[0,1],~α,β>0$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $[t,1]$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel-kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval $(-a,a),a>0,$ is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight $(1-x^2)^β, x\in[-1,1]$.
We analyze the eigenvalue density for the Laguerre and Jacobi $β$-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms $o(1)$, in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian $β$-ensemble, and Laguerre $β$-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.
In one dimensional transport problems the scattering matrix $S$ is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends. For $S$ a random unitary matrix, the singular value probability distribution function of these blocks is calculated. The same is done when $S$ is constrained to be symmetric, or to be self dual quaternion real, or when $S$ has real elements, or has real quaternion elements. Three methods are used: metric forms; a variant of the Ingham-Seigel matrix integral; and a theorem specifying the Jacobi random matrix ensemble in terms…
Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $β$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $β$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing result…
In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $α=β$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $α=-1/2,$ and $α=1/2$ and the (shift…
In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^α{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~α,t>0. \end{equation*} By using the ladder operator approach, we obtain that an auxiliary quantity $R_n(t)$, which is closely related to the recurrence coefficients of monic polynomials orthogonal with $w(z)$, satisfies a particular Painlevé V equation.
The $β$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $β$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and al…
This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo-Miwa-Okamoto $σ$-form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced …
Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $β$ generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices $β= 1,2$ and 4, and also to the spectral density in the corresponding $β$-ensemble w…
In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval $(-a,a)\:(0<a<1)$ is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by $H_{n}(a)$, $R_{n}(a)$ and $r_{n}(a)$. We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable $a…
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…
A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the defin…
For a beta-Jacobi ensemble determined by parameters a_1, a_2 and n, under the restriction that the three parameters go to infinity with n and a_1 being of small orders of a_2, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.
We study Muttalib--Borodin ensembles --- particular eigenvalue PDFs on the half-line --- with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib--Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib--Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetr…
The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second order second degree ODEs which are related to certain \mbox{Painlevé-VI} transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to \mbox{Painlevé-V} transcendents. Using transformations relating the second degr…
We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.
We study the Jacobi unitary ensemble perturbed by an algebraic singularity at $t>1$. For fixed $t$, this is the modified Jacobi ensemble studied by Kuijlaars {\it{et al.}} The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely $t\to 1^+$. In the double scaling limit case when $t- 1$ is of the order of magnitude of $1/n^2$, $n$ being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge $1$, described by the $ψ$-functions for a certain second-order nonlinear equation. The equation is r…
We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.
Inspired by Aomoto's $q$-Selberg integral, the orthogonal ensemble in the exponential lattice is considered in this paper. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus, corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. Examples including Al-Salam & Carlitz, $q$-Laguerre, little $q$-Jacobi and big $q$-Jacobi cases are considered.
We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor …
In this paper we prove that, after an appropriate rescaling, the sum of moments $\mathbb{E}_{N}^{(s)} \left( Tr \left( |\mathbf{H}|^{2k+2}+|\mathbf{H}|^{2k}\right) \right)$ of an $N\times N$ Hermitian matrix $\mathbf{H}$ sampled according to the generalized Cauchy (also known as Hua-Pickrell) ensemble with parameter $s>0$ is a continuous-Hahn polynomial in the variable $k$. This completes the picture of the investigation that began by Cunden, Mezzadri, O'Connell and Simm who obtained analogous results for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Ja…
We compute analytically the probability density function (pdf) of the largest eigenvalue $λ_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($β= 2$), orthogonal ($β=1$) and symplectic ($β=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $λ_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $β$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, …
The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding joint probability density function of its eigenvalues was identified. Moreover a relation with the Cauchy two-matrix model was discovered but never thoroughly investigated, leaving open in particular the following question: How are the kernels of the Pfaffian point process of the Bures random matrix ensemble related to the ones of the determinantal point process of the Cauchy two-matrix model and moreo…
The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our stud…
Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le ...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=κ\cdot (a,b,1)$ with $a,b>0$ fixed, we derive a central limit theorem for the distributions above for $κ\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomi…
The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom $ [0,s) $ or at the top $ (s,1] $, utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables $s,t$. To this end we develop new general results about Cauchy …
Let $\mathbf{W}_1$ and $\mathbf{W}_2$ be independent $n\times n$ complex central Wishart matrices with $m_1$ and $m_2$ degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices $(\mathbf{W}_1+\mathbf{W}_2)^{-1}\mathbf{W}_1$, which are analogous to those of F matrices ${\bf W}_1 {\bf W}_2^{-1}$ and those of the Jacobi unitary ensemble (JUE). Defining $α_1=m_1-n$ and $α_2=m_2-n$, we derive new exact distribution formulas in terms of $(α_1+α_2)$-dimensional matrix determinants, with elements involving derivatives of Legendre poly…
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.
The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $β$-Ensemble (C$β$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$β$…
We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{λ_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $τ_n=<\sum_{i=1}^Nλ_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming lead…
We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for…
Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{δ,β}^{(n)} \prod_{1\leq k<l\leq n}| e^{\iiθ_k}-e^{\iiθ_l}|^β\prod_{j=1}^{n}(1-e^{-\iiθ_j})^δ (1-e^{\iiθ_j})^{\overlineδ} $$ with $\Re δ> -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(α_0, ..., α_{n-1})$. We introduce here a deformation $(γ_0, >..., γ_{n-1})$ of these coeff…
According to the classification scheme of the generalized random matrix ensembles, we present various kinds of concrete examples of the generalized ensemble, and derive their joint density functions in an unified way by one simple formula which was proved in [2]. Particular cases of these examples include Gaussian ensemble, chiral ensemble, new transfer matrix ensembles, circular ensemble, Jacobi ensembles, and so on. The associated integration formulae are also given, which are just many classical integration formulae or their variation forms.
We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.
We apply the method of determinants to study the distribution of the largest singular values of large $ m \times n $ real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor of $ \frac{1}{m^2\*n^2}$)largest singular values agree in the limit with the statistics of the inhomogeneous Poisson random point process with the intensity $ \frac{1}π x^{-3/2} $ and, therefore, are different from the Tracy-Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the…
Examples of the $β$-Jacobi ensemble specify the joint distribution of the transmission eigenvalues in scattering problems. In this context, there has been interest in the distribution of the trace, as the trace corresponds to the conductance. Earlier, in the case $β= 1$, the trace statistic was isolated in studies of covariance matrices in multivariate statistics, where it is referred to as Pillai's $V$ statistic. In this context, Davis showed that for $β= 1$ the trace statistic, and its Fourier-Laplace transform, can be characterised by $(N+1) \times (N+1)$ matrix differential equations. For …
Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_c=UL is a monic generalized Jacobi matrix associated with the function F_c(z)=zF(z)+1. It turns out that the Christoffel transformation J_c of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at infinity of the poles of the Padé approximants of the function F_c although F_c is holomorphic at infinity. The case of the UL-fac…
Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary ensembles are recovered. At the same time, we study the perturbation invariability of the level densities of random matrix unitary ensembles. After the weight function associated with the 1-level correlation function is appended a polynomial multiplicative factor, the level density is invariant in the weak sense.
This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ-function of the CKP hiearc…
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, diff…
Using a Coulomb gas technique, we compute analytically the probability $\mathcal{P}_β^{(C)}(N_+,N)$ that a large $N\times N$ Cauchy random matrix has $N_+$ positive eigenvalues, where $N_+$ is called the index of the ensemble. We show that this probability scales for large $N$ as $\mathcal{P}_β^{(C)}(N_+,N)\approx \exp\left[-βN^2 ψ_C(N_+/N)\right]$, where $β$ is the Dyson index of the ensemble. The rate function $ψ_C(κ)$ is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate function, around its minimum at $κ…
The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g.~condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg int…
