Interrelations of discrete Painlevé equations through limiting procedures

We study the discrete Painlevé equations associated to the affine Weyl group which can be obtained by the implementation of a special limits of -associated equations. This study is motivated by the existence of two -associated discrete both having a double ternary dependence in their coefficients and which have not been related before. We show here that two equations correspond to two different limits of a -associated discrete Painlevé equation. Applying the same limiting procedures to other -associated equations we obtained several -related equations most of which have not been previously derived.     

Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.     

Emptiness Formation Probability of the Six-Vertex Model and the Sixth Painlevé Equation

We show that the emptiness formation probability of the six-vertex model with domain wall boundary conditions at its free-fermion point is a \({\tau}\)-function of the sixth Painlevé equation. Using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit.     

Hankel Determinant and Orthogonal Polynomials for a Gaussian Weight with a Discontinuity at the Edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.     

Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles

In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \) -form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\) , \(R_n(a)\) and \(r_n(a)\) . We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \) -form and Painlevé IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\) .