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 definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.
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.
We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses either one soft edge and one hard edge (Laguerre-type) or two hard edges (Jacobi-type). We also present some applications in the theory of random matrices. In particular, we can deduce from our results asymptotics for partition functions with singularities, central limit theorems, correlations of the characteristic polynomials, and gap probabilities for (piecewi…
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…
We define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by Pal-Shkolnikov. One consequence is the construction of particle systems with local interactions whose fixed time d…
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi 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.
We show that for $β\ge 1$ the semigroups of $β$-Laguerre and $β$-Jacobi processes of different dimensions are intertwined in analogy to a similar result for $β$-Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $β\ge 1$ the ones obtained for $β=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key step towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving certain Gibbs measures invariant. Finally as a by product we obtain a relation betw…
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates f…
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be …
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual stat…
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree $\ell=1,2,...$, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change.
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner …
We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-Pöschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by…
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 derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalizatio…
We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one dimensional quantum mechanics and the corresponding X_{\ell} Laguerre and Jacobi polynomials (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417). The new X_{\ell} Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X_{\ell} Jacobi polynomials and the potentials, whereas the known X_{\ell} L…
In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.
The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involv…
We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape in…
Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these para…
Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS.…
In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relation…
The ensemble average of $| \sum_{j=1}^N e^{i k λ_j} |^2$ is of interest as a probe of quantum chaos, as is its connected part, the structure function. Plotting this average for model systems of chaotic spectra reveals what has been termed a dip-ramp-plateau shape. Generalising earlier work of Brézin and Hikami for the Gaussian unitary ensemble, it is shown how the average in the case of the Laguerre unitary ensemble can be reduced to an expression involving the spectral density of the Jacobi unitary ensemble. This facilitates studying the large $N$ limit, and so quantifying the dip-ramp-platea…
Coefficients in the expansions of the form $\partial P_{n}(λ;z)}/\partialλ=\sum_{k=0}^{n}a_{nk}(λ)P_{k}(λ;z)$, where $P_{n}(λ;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi) orthogonal polynomial of variable $z$ and $λ$ is a parameter, are evaluated. A method we adopt in the present paper differs from that used by Fröhlich [Integral Transforms Spec. Funct. 2 (1994) 253] for the Jacobi polynomials and by Koepf [Integral Transforms Spec. Funct. 5 (1997) 69] for the generalized Laguerre and the Gegenbauer polynomials.
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\p_{n}\}_{n=0}^{\infty}$, with $\text{deg}\,p_{n}=n$ that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences $\{p_{n}\}_{n=1}^{\infty}$, $\text{deg}\,p_{n}=n$ (without the cons…
We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define "Hamiltonians" enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the so…
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 develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre and Jacobi ensembles for all the symmetry classes beta = 1,2, 4 and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well su…
New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial $g$. The cases where $g$ is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials…
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…
Muttalib--Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod_{1 \le j < k \le N}(λ_k - λ_j) (λ_k^θ- λ_j^θ)$. We study the Laguerre and Jacobi versions of this model --- so named by the form of the one-body interaction terms --- and show that for $θ\in \mathbb Z^+$ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general $θ> 0$, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent …
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.
$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $β$-Hermite, $β$-Laguerre, and $β$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $β\to\infty$ in the $β$-Hermite and $β$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $Σ_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $Σ_N^{-1}$ with formulas for the eigenvalues and eigenvectors of …
Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $α_{n}=ka_{n}$, $β_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial sequence defined by the three-term recurrence \[ α_{0}P_{1}(x)+(β_{0}-x)P_{0}(x)=0,\ α_{n}P_{n+1}(x)+(β_{n}-x)P_{n}(x)+α_{n-1}P_{n-1}(x)=0 \] for $n\geq1$, with $P_{0}(x)=1$. Let $J$ be the corresponding Jacobi (tridiagonal) matrix, i.e. $J_{n,n}=β_{n}$, $J_{n,n+1}=J_{n+1,n}=α_{n}$ for $n\geq0$. Then $J^{-1}$ exists and belongs to the trace class. We derive an expli…
Field-Modified Quantum Potentials from Tridiagonal Representations: Analytical Spectra and Galerkin Simulations
Khalifa University of Science and Technology（哈利法科学技术大学）；Center for Cyber-Physical Systems (C2PS), Khalifa University of Science and Technology（哈利法科学技术大学网络物理系统中心）
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…
We derive exact results for gap probabilities, as well as densities of extreme eigenvalues for six complex random matrix ensembles of fundamental importance. These are Gauss-Wigner, Laguerre-Wishart, Cauchy-Lorentz (two variants), Jacobi-MANOVA and Bures-Hall (trace unrestricted). We deal with both correlated and uncorrelated cases. Extensive Monte-Carlo simulations based on explicit matrix models or Dyson's log-gas formalism have also been performed, which corroborate all analytical results.
The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal polynomials of a single variable.
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.
We present a derivation of classical Hermite, Laguerre, and Jacobi orthogonal polynomials directly through the Gram-Schmidt orthogonization process. The derivation uses certain generalized Vandermonde determinants with entries defined by Gamma and Beta functions. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.
Okamoto has obtained a sequence of $τ$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multi-dimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, …
