Symmetric Sextic Freud Weight

Abstract

This paper investigates properties of the sequence of coefficients $({\beta }_n{)}_{n\ge 0}$ in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight $$\omega (x;\tau ,t)=\exp (-x^6+\tau x^4+t{x}^2),x\in \mathbb{R},$$ with real parameters $\tau$ and $t$. It is known that the recurrence coefficients ${\beta }_n$ satisfy a fourth-order nonlinear discrete equation, which is a special case of the second member of the discrete Painlevé I hierarchy, often known as the ''string equation''. The recurrence coefficients have been studied in the context of Hermitian one-matrix models and random symmetric matrix ensembles with researchers in the 1990s observing ''chaotic, pseudo-oscillatory'' behaviour. More recently, this ''chaotic phase'' was described as a dispersive shockwave in a hydrodynamic chain. Our emphasis is a comprehensive study of the behaviour of the recurrence coefficients as the parameters $\tau$ and $t$ vary. Extensive computational analysis is carried out, using Maple, for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases in terms of generalised hypergeometric functions and modified Bessel functions. The results highlight the rich algebraic and analytic structures underlying the Freud weight and its connections to integrable systems.

55 pages, 28 figures

References in corpus (24)

• Orthogonal polynomial ensembles in probability theory
• Normal random matrix ensemble as a growth problem
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• A holomorphic and background independent partition function for matrix models and topological strings
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• A Generalized Freud Weight
• Properties of Generalized Freud Polynomials
• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• Asymptotics for the partition function in two-cut random matrix models
• Bi-rational maps in four dimensions with two invariants
• Asymptotic expansion of beta matrix models in the multi-cut regime
• On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
• A Generalised Sextic Freud Weight
• Generalised Airy Polynomials
• Thermodynamic Limit and Dispersive Regularisation in Matrix Models
• Generalised higher-order Freud weights
• Multiple Scale Asymptotics of Map Enumeration
• Integrable maps in 4D and modified Volterra lattices
• Special solutions of a discrete Painlevé equation for quantum minimal surfaces
• Generalised Freud's equation and level densities with polynomial potential