Asymptotics of the Hankel determinant and orthogonal polynomials arising from the information theory of MIMO systems
Chao Min, Xiaoqing Wu
TL;DR
The paper studies the Hankel determinant $D_n(t)$ and associated orthogonal polynomials for the deformed Laguerre weight $w(x;t)=x^{\alpha}e^{-x}(x+t)^{\lambda}$, motivated by MIMO information theory. It combines a ladder-operator framework with Coulomb-fluid (Dyson) analysis to derive discrete and differential-difference equations for the recurrence coefficients $\alpha_n(t)$ and $\beta_n(t)$, a second-order ODE for the monic polynomials $P_n(x;t)$, and relations among $D_n(t)$, $p(n,t)$, and $H_n(t)$. The authors obtain rigorous large-$n$ asymptotics for $\alpha_n(t)$, $\beta_n(t)$, $p(n,t)$, $H_n(t)$, $D_n(t)$, and $h_n(t)$, including explicit leading terms and higher-order corrections; the results accommodate $\lambda=0$, recovering the classical Laguerre case via Barnes $G$-function constants. They also derive long-time asymptotics $t\to\infty$ with fixed $n$, giving precise expansions for the recurrence coefficients, the Hankel determinant, and the normalized constants, and highlighting connections to Painlevé V and random matrix theory. These contributions advance the analytic understanding of semi-classical Laguerre-type systems arising in information theory and mathematical physics.
Abstract
We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $$ w(x; t) = {x^α}{\mathrm e^{ - x}}{(x + t)^λ},\qquad x\in \mathbb{R}^{+}, $$ with parameters $α> -1,\; t > 0$ and $λ\in \mathbb{R}$. This problem originates from the information theory of single-user multiple-input multiple-output (MIMO) systems studied by Chen and McKay [{\em IEEE Trans. Inf. Theory} {\bf 58} ({2012}) {4594--4634}]. By using the ladder operators for orthogonal polynomials with general Laguerre-type weights, we obtain a system of difference equations and a system of differential-difference equations for the recurrence coefficients $α_n(t)$ and $β_n(t)$. We also show that the orthogonal polynomials satisfy a second-order ordinary differential equation. By using Dyson's Coulomb fluid approach, we obtain the large $n$ asymptotic expansions of the recurrence coefficients $α_n(t)$ and $β_n(t)$, the sub-leading coefficient $\mathrm p(n, t)$ of the monic orthogonal polynomials, the Hankel determinant $D_n(t)$ and the normalized constant $h_n(t)$ for fixed $t\in\mathbb{R}^{+}$. We also discuss the long-time asymptotics of these quantities as $t\rightarrow\infty$ for fixed $n\in\mathbb{N}$.