Gaussian unitary ensembles with two jump discontinuities,PDEs, and the coupled Painlevé II and IV systems

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in Wu and Xu [arXiv: 2002.11240v2] by a study of the Riemann‐Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as , the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second‐order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.     

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.     

Perturbed Laguerre unitary ensembles,Hankel determinants,and information theory

This article investigates a key information‐theoretic performance metric in multiple‐antenna wireless communications, the so‐called outage probability. The article is partly a review, with the methodology based mainly on [10], while also presenting some new results. The outage probability may be expressed in terms of a moment generating function, which involves a Hankel determinant generated from a perturbed Laguerre weight. For this Hankel determinant, we present two separate integral representations, both involving solutions to certain non‐linear differential equations. In the second case, this is identified with a particular σ‐form of Painlevé V. As an alternative to the Painlevé V, we show that this second integral representation may also be expressed in terms of a non‐linear second order difference equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Painlevé VI,Painlevé III,and the Hankel determinant associated with a degenerate Jacobi unitary ensemble

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 to the Jimbo–Miwa–Okamoto σ-form of the Painlevé III. In the end, we obtain the asymptotic behavior of the Hankel determinant as t→1⁻ and t→0⁺ in two important cases, respectively.     

Painlevé VI and the Unitary Jacobi Ensembles

The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well‐known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin‐spin correlation functions of the two‐dimensional Ising model in the work of McCoy, Tracy, and Wu. The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double‐Wishart distribution. We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ‐form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .     

Affine Weyl group symmetries of Frobenius Painlevé equations

In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.     

Solution of the equivalence problem for the Painlevé IV equation

We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, a = b = 0; only one parameter is zero, b = 0; and the parameter b ?? 0. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

On the question of representation of ultrafilters in a product of measurable spaces

We consider nonlinear ordinary differential equations up to the sixth order that are associated with the heat equation. Each of them is subjected to the Painlevé analysis. For the fourth- and sixth-order equations we obtain a criterion for having the Painlevé property; for the fifth-order equation we formulate necessary conditions for passing the Painlevé test. We also present a fifth-order equation analogous to the Chazy-3 equation.     

Quicksilver Solutions of a q‐Difference First Painlevé Equation

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qP_I), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qP_I, is also applicable to other q‐difference Painlevé equations.     

Painlevé XXXIV Asymptotics of Orthogonal Polynomials for the Gaussian Weight with a Jump at the Edge

We study the uniform asymptotics of the polynomials orthogonal with respect to analytic weights with jump discontinuities on the real axis, and the influence of the discontinuities on the asymptotic behavior of the recurrence coefficients. The Riemann–Hilbert approach, also termed the Deift–Zhou steepest descent method, is used to derive the asymptotic results. We take as an example the perturbed Gaussian weight , where θ(x) takes the value of 1 for x < 0 , and a nonnegative complex constant ω elsewhere, and as . That is, the jump occurs at the edge of the support of the equilibrium measure. The derivation is carried out in the sense of a double scaling limit, namely, and . A crucial local parametrix at the edge point where the jump occurs is constructed out of a special solution of the Painlevé XXXIV equation. As a main result, we prove asymptotic formulas of the recurrence coefficients in terms of a special Painlevé XXXIV transcendent under the double scaling limit. The special thirty‐fourth Painlevé transcendent is shown free of poles on the real axis. A consistency check is made with the reduced case when ω= 1 , namely the Gaussian weight: the polynomials in this case are the classical Hermite polynomials. A comparison is also made of the asymptotic results for the recurrence coefficients between the case when the jump happens at the edge and the case with jump inside the support of the equilibrium measure. The comparison provides a formal asymptotic approximation of the Painlevé XXXIV transcendent at positive infinity.     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree

In this paper we construct all Painlevé-type differential equations of the form (d²y/dx²)² = F(x,y,dy/dx), where F is rational in y and y′=dy/dx, locally analytic in x, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear in y″ in the class of equations under investigation forces F to be a polynomial in y and y′. We find exactly six distinct classes of second-degree Painlevé equations, denoted SD-I,??,SD-VI, some of which further subdivide into canonical subcases. Only the first three classes (or at least equations transformable to the first three classes) and part of the sixth have appeared previously in the literature, especially the work of Chazy and Bureau. The fourth and fifth classes are new. The unified treatment of SD-I, which we call the “master Painlevé equation,” is new. Complete solutions are given in terms of the classical Painlevé transcendents, elliptic functions, or solutions of linear equations. In an appendix, it is shown that a class of second-degree equations generalizing the Appell equation can always be reduced to a second-order linear equation.     

Differential equations for deformed Laguerre polynomials

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painlevé transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painlevé equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories by showing how either can be used to obtain a characterization of a more general Laguerre unitary ensemble average in terms of the Hamiltonian system for Painlevé V.     

λ‐symmetries,nonlocal transformations and first integrals to a class of Painlevé–Gambier equations

In this work, we consider a class of Painlevé–Gambier equations that model the motion of chain ball drawing with constant force in the frictionless surface. λ‐symmetries, first integrals, integrating factors, nonlocal transformations and local transformations are derived by using the some recent studies that are proposed by Muriel and Romero. Copyright © 2012 John Wiley & Sons, Ltd.     

Antiquantization of deformed Heun-class equations

We consider deformed Heun-class equations, i.e., equations of the Heun class with an added apparent singularity. We prove that each deformed Heun-class equation under antiquantization realizes a transfer from the Heun-class equation to the corresponding Painlevé equation, and we completely list such transfers.     

Exact Solutions of Fisher and Generalized Fisher Equations with Variable Coefficients

In this work,we consider a Fisher and generalized Fisher equations with variable coefficients.Usingtruncated Painlevé expansions of these equations,we obtain exact solutions of these equations with a constrainton the coefficients a(t)and b(t).     

All Binomial-Type Painlevé Equations of the Second Order and Degree Three or Higher

In this paper we construct all rational Painlevé-type differential equations which take the binomial form, (d²y/dx²)ⁿ = F(x,y,dy/dx), where n ≥ 3, the case n = 2 having previously been treated in Cosgrove and Scoufis [1]. While F is assumed to be rational in the complex variables y and y′ and locally analytic in x, it is shown that the Painlevé property together with the absence of intermediate powers of y″ forces F to be a polynomial in y and y′. In addition to the six classes of second-degree equations found in the aforementioned paper, we find nine classes of higher-degree binomial Painlevé equations, denoted BP-VII,..., BP-XV, of which the first seven are new. Two of these equations are of the third degree, two of the fourth degree, three of the sixth degree, and two of arbitrary degree n. All equations are solved in terms of the first, second or fourth Painlevé transcendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second-order nth equations (not necessarily of Painlevé type) which can also be solved in terms of Painlevé transcendents or elliptic functions.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.