In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlevé equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlevé equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painlevé equations.
In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of $$w(x,t)=(x+t)^λx^αe^{-x}$$ with $ x\geq 0,\ t>0,\ α>0,\ α+λ+1 > 0.$ The Deift-Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling $s=4nt,$ $n\to \infty,$ $t\to 0 $ such that $s$ is positive and finite, at the hard edge, the limiting kernel can be described by the $φ$-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P$_…
The scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles is evaluated in terms of a Painlevé V transcendent. This same Painlevé V transcendent is known from the work of Tracy and Widom, where it has been shown to specify the scaled distribution of the smallest eigenvalue in the Laguerre unitary ensemble. The starting point for our calculation is the scaled $k$-point distribution of every odd labelled eigenvalue in two superimposed Laguerre orthogonal ensembles.
In this paper, we study Hankel determinants generated from a perturbed Laguerre weight function, Under the double scaling scheme, we give the uniform asymptotic approximations of Hankel determinants in terms of a solution of a third-order nonlinear differential equation, which is equivalent to a particular Painleve V equation. In fact, this Painleve V equation is equivalent to the general Painleve III equation. The asymptotic approximations of the leading coefficients and the recurrence coefficients for the corresponding orthogonal polynomials also involve the painleve V equation. The asymptot…
We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.
Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
International Chair of Mathematical Physics and Applications (ICMPA–UNESCO Chair)（国际数学物理与应用 chair（ICMPA-UNESCO 讲席））；University of Abomey-Calavi（阿波美卡拉维大学）
We consider an ensemble which interpolates the Laguerre orthogonal ensemble and the Laguerre symplectic ensemble. This interpolating ensemble was introduced earlier by the author and Rains in connection with a last passage percolation model with a symmetry condition. In this paper, we obtain a Painelevé V expression for the distribution of the rightmost particle of the interpolating ensemble. Special cases of this result yield the Painlevé V expressions for the largest eigenvalues of Laguerre orthogonal ensemble and Laguerre symplectic ensemble of finite size.
We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.
Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_{6}^{(1)}$ Sakai surface
The rational solutions for the discrete Painlevé II equation are constructed based on the bilinear formalism. It is shown that they are expressed by the determinant whose entries are given by the Laguerre polynomials. Continuous limit to the Devisme polynomial representation of the rational solutions for the Painlevé II equation is also discussed.
A determinant expression for the rational solutions of the Painlevé III (P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given. Degeneration of this determinant expression to that for the rational solutions of P$_{\rm II}$ is discussed by applying the coalescence procedure.
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 …
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…
This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) γ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two differ…
A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. They are: the Laguerre polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials, the Ohyama polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as…
With $<\cdot>$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,μ) := < \prod_{l=1}^N χ_{(0,\infty)\backslash I}^{(l)} (λ- λ_l)^μ>$ for $I = (0,s)$ and $I = (s,\infty)$, where $χ_I^{(l)} = 1$ for $ λ_l \in I$ and $χ_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that $\tilde{E}_N(I;a,μ)$ is a $τ$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, …
In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrenc…
We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels à la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlevé III equation related to the Fredholm determinant.
We review the construction of the mixed Painlevé P$_{III-V}$ system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P$_{III}$ or P$_{V}$. The aim of this paper is to describe solutions of P$_{III-V}$ model. In particular, we determine and classify rational, power series and transcendental solutions of P$_{III-V}$. A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P$_{III-V}$ e…
On the discrete Painlevé equivalence problem, non-conjugate translations and nodal curves
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用研究所）；Institute of Mathematics, University of Warsaw（华沙大学数学研究所）；Waseda Institute for Advanced Study, Waseda University（早稻田大学高等研究院）
We consider the 2-dimensional Toda lattice tau functions $τ_n(t,s;η,θ)$ deforming the probabilities $τ_n(η,θ)$ that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc $(η,θ)$ of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painleve VI equation.
We consider the squared singular values of the product of $M$ standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov \cite{St14} that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by …
