Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics

Abstract

We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$w(x;t):={\mathrm{e}}^{-x^2}{\left(1+t{x}^2\right)}^{\lambda },x\in \mathbb{R},$$ where $t>0,\lambda \in \mathbb{R}$. This weight is related to the single-user MIMO systems in information theory. It is shown that the recurrence coefficient ${\beta }_n(t)$ is related to a particular Painlevé V transcendent, and the sub-leading coefficient $\mathrm{p}(n,t)$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by ${\beta }_n(t)$ and $\mathrm{p}(n,t)$, respectively. This enables us to obtain the large $n$ full asymptotic expansions for ${\beta }_n(t)$ and $\mathrm{p}(n,t)$ with the aid of Dyson's Coulomb fluid approach. We also consider the Hankel determinant $D_n(t)$, generated by the perturbed Gaussian weight. It is found that $H_n(t)$, a quantity allied to the logarithmic derivative of $D_n(t)$, can be expressed in terms of ${\beta }_n(t)$ and $\mathrm{p}(n,t)$. Based on this result, we obtain the large $n$ asymptotic expansion of $H_n(t)$ and then that of the Hankel determinant $D_n(t)$.

28 pages

References in corpus (3)

• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight

Cited by in corpus (1)

• Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators