Painlevé V and confluent Heun equations associated with a perturbed Gaussian unitary ensemble

Abstract

We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w(z,t)=|z|α(z2+t)λe−z2,z∈R,t>0,α>−1,λ>0, which arises from a symmetrization of a semi-classical Laguerre weight wLag(z,t)=zγ(z+t)ρe−z,z∈R+,t>0,γ>−1,ρ>0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., PVλ22,−(1−(−1)nα)28,−2n+α+2λ+12,−12. We also consider the quantity σn(t)≔2tddtlnDn(t), which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson’s Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation.