Multicritical points of unitary matrix model with logarithmic potential identified with Argyres–Douglas points

In our recent publications, the partition function of the Gross–Witten–Wadia unitary matrix model with the logarithmic term has been identified with the [Formula: see text] function of a certain Painlevé system, and the double scaling limit of the associated discrete Painlevé equation to the critical point provides us with the Painlevé II equation. This limit captures the critical behavior of the [Formula: see text], [Formula: see text], [Formula: see text] supersymmetric gauge theory around its Argyres–Douglas 4D superconformal point. Here, we consider further extension of the model that contains the [Formula: see text]th multicritical point and that is to be identified with [Formula: see text] theory. In the [Formula: see text] case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model.

$q$ -Racah Ensemble and Discrete Painlevé Equation

The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

On properties of a deformed Freud weight

We study the recurrence coefficients of the monic polynomials [Formula: see text] orthogonal with respect to the deformed (also called semi-classical) Freud weight [Formula: see text] with parameters [Formula: see text]. We show that the recurrence coefficients [Formula: see text] satisfy the first discrete Painlevé equation (denoted by d[Formula: see text]), a differential–difference equation and a second-order nonlinear ordinary differential equation (ODE) in [Formula: see text]. Here [Formula: see text] is the order of the Hankel matrix generated by [Formula: see text]. We describe the asymptotic behavior of the recurrence coefficients in three situations: (i) [Formula: see text], [Formula: see text] finite, (ii) [Formula: see text], [Formula: see text] finite, (iii) [Formula: see text], such that the radio [Formula: see text] is bounded away from [Formula: see text] and closed to [Formula: see text]. We also investigate the existence and uniqueness for the positive solutions of the d[Formula: see text]. Furthermore, we derive, using the ladder operator approach, a second-order linear ODE satisfied by the polynomials [Formula: see text]. It is found as [Formula: see text], the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, [Formula: see text], associated with [Formula: see text] when [Formula: see text] tends to infinity.

Stokes phenomena in discrete Painlevé I

In this study, we consider the asymptotic behaviour of the first discrete Painlevé equation in the limit as the independent variable becomes large. Using an asymptotic series expansion, we identify two types of solutions which are pole-free within some sector of the complex plane containing the positive real axis. Using exponential asymptotic techniques, we determine Stokes phenomena effects present within these solutions, and hence the regions in which the asymptotic series expression is valid. From a careful analysis of the switching behaviour across Stokes lines, we find that the first type of solution is uniquely defined, while the second type contains two free parameters, and that the region of validity may be extended for appropriate choice of these parameters.