In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $ (0,s) $ at the hard edge contains $ k $ eigenvalues, was evaluated in terms of a Painlevé V transcendent in $ σ$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $ s $ behaviour of the Painlevé V equation in $ σ$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlevé \IIId transcendent in $ σ$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function.
This is the last part of a series of three papers entitled "Four-dimensional Painlevé-type equations associated with ramified linear equations". In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlevé-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system.
We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by …
In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier pa…
The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last …
We continue with the study of the Hankel determinant, $$ D_{n}(t,α):=\det\left(\int_{0}^{\infty}x^{j+k}w(x;t,α)dx\right)_{j,k=0 }^{n-1}, $$ generated by singularly perturbed Laguerre weight, $$ w(x;t,α):=x^α{\rm e}^{-x}\:{\rm e}^{-t/x}, \quad 0\leq x<\infty,\;\;\;α>0,\;\;\;\;t>0, $$ obtained through a deformation of the Laguerre weight function, $$ w(x;0,α):=x^α{\rm e}^{-x},\quad 0\leq x<\infty,\;\; α>0, $$ via the multiplicative factor ${\rm e}^{-t/x}$. \\ An earlier investigation was made on the finite $n$ aspect of the problem, this has appeared in \cite{ci1}. There, it was found that the l…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlevé VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of Bäcklund transformations. For particular lines in the lattice, this proves conjectures of Bazhanov and Mangazeev. As applications, we describe the behaviour of the corresponding solutions near the singular points of Painlevé VI, and obtain several new properties of our polynomials.
We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierar…
Higher dimensional analogs of the Painlevé equations have been proposed from various aspects. In recent studies, 4-dimensional analogs of the Painlevé equations were classified into 40 types. The aim of the present paper is to geometrically characterize these 40 types of equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. We obtain two functionally independent conserved quantities $H_1$ and $H_2$ for each system. We construct fibrations whose fiber at a general point $h_i $ is the spectral curve of the system with $H_i=h_i$ f…
In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$ 'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an in…
We elaborate on a recently conjectured relation of Painlevé transcendents and 2D CFT. General solutions of Painlevé VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton partition functions in $\mathcal{N}=2$ supersymmetric gauge theories with $N_f=0,1,2,3,4$. Resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toe…
Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation
We consider a $q$-Painlevé III equation and a $q$-Painlevé II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $τ$ functions.
All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as $n\in\mathbb{Z}$ increases and how the patterns vary with $m\in\mathbb{C}$. This study suggests that it is reasonable to consider the rational solutions in the li…
In this paper some open problems for Painlevé equations are discussed. In particular the following open problems are described: (i) the Painlevé equivalence problem; (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé equations; and (iv) the classification of properties of Painlevé equations.
In this paper, we study the Painlevé VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $λ\left( t\right) $ is unitary, then $λ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order $k$, we establish the existence of a limiting profile of the rogue wave in the large-$k$ limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables --- the rogue wav…
The Painlevé-III equation with parameters $Θ_0=n+m$ and $Θ_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with …
In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^μ\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^μ(1+1/z)^ν\exp(tz) $ -- known $τ$-functions for the \PIIIa and \PV systems -- are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlevé equations associated with the degenerations of the rational surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlevé {\rm II}) and $ D^{(1)}_{5} \to E^{(1)}_{6}…
It is known that all $τ$ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions explicitly by using combinatorics. In this paper, we show the convergence of the formal series, including the solutions of more general equations. And by the absolute convergence of $τ$ series, the convergence of the conformal block function ($c=1$) also follows since it is a partial sum of the $τ$ series. We also characterized the form of a homogeneous quadratic equation with a…
We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlevé III equation. Next we consider the q-Laguerre or generalized Stieltjes-Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlevé V eq…
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…
In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{ρ_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev…
We review some aspects of recent work concerning double scaling limits of singularly perturbed hermitian random matrix models and their connection to Painlevé equations. We present new results showing how a Painlevé III hierarchy recently proposed by the author can be connected to the Lenard recursion formula used to construct the Painlevé I and II hierarchies.
Modified Painlevé systems with meromorphic solutions for polynomial Hamiltonians of all degrees
Faculty of Mathematics Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学与机械学院）；School of Mathematics and Physics, University of Portsmouth（朴茨茅斯大学数学与物理学院）
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting…
The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé \IIId system and its associated linear isomonodromic system characterise the scaled distribution. We un…
A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $τ$-function of the fifth Painlevé system. This $τ$-function, $ τ(s) $, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $τ$-function is of the separatrix, or partially truncated type.
We study a family of Liénard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this equations family. To this end we use an approach, which is bases on application of nonlocal transformations. By studying connections between this family of Liénard--type equations and type III Painlevé--Gambier equations, we obtain four new integrability criteria. We illustrate our results by…
The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions ${}_0 F_M$, also referred to as hyper-Bessel functions. In the case $M=1$ it is well known that the corresponding gap probability for no squared singular values in $(0,s)$ can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach t…
