Asymptotics of the Hankel determinant and orthogonal polynomials arising from the information theory of MIMO systems

Abstract

We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $w(x;t)=x^{\alpha }{\mathrm{e}}^{-x}(x+t{)}^{\lambda },x\in {\mathbb{R}}^{+}$ with parameters $\alpha >-1,t>0$ and $\lambda \in \mathbb{R}$. 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 ${\alpha }_n(t)$ and ${\beta }_n(t)$. 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 ${\alpha }_n(t)$ and ${\beta }_n(t)$, 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 $t\to \infty$ for fixed $n\in \mathbb{N}$. The large $n$ and large $t$ asymptotics of the above quantities are very important for the study of the asymptotics of the mutual information distribution and two fundamental quantities (the outage capacity and the error probability) for single-user MIMO systems.