On the asymptotics of some large Hankel determinants generated by Fisher–Hartwig symbols defined on the real line

Abstract

We investigate the asymptotics of Hankel determinants of the form detj,k=0N−1[∫ΩdxωN(x)∏i=1m∣μi−x∣2qixj+k] as N→∞ with q and μ fixed, where Ω is an infinite subinterval of R and ωN(x) is a positive weight on Ω. Such objects are natural analogs of Toeplitz determinants generated by Fisher–Hartwig symbols, and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials. The reduced density matrices of certain one-dimensional systems of trapped impenetrable bosons can also be expressed in terms of Hankel determinants of this form. We focus on the specific cases of scaled Hermite and Laguerre weights. We compute the asymptotics by using a duality formula expressing the N×N Hankel determinant as a 2(q1+⋯+qm)-fold integral, which is valid when each qi is natural. We thus verify, for such q, a recent conjecture of Forrester and Frankel derived using a log-gas argument.