Applications and generalizations of Fisher-Hartwig asymptotics

TL;DR

This paper explores the Fisher-Hartwig asymptotics for Toeplitz and Hankel determinants, applying it to physical models like the Ising model and Bose gas, and generalizing it to random matrix ensembles.

Contribution

It introduces new applications of Fisher-Hartwig asymptotics to Ising correlations and generalizes the formula to Hankel determinants and random matrix averages.

Findings

Derived asymptotic decay of Ising correlations above T_c.

Generalized Fisher-Hartwig formula to Hankel determinants.

Extended asymptotic analysis to random matrix ensembles.

Abstract

Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants.