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

Abstract

We investigate the asymptotics of the determinant of N by N Hankel matrices generated by Fisher-Hartwig symbols defined on the real line, as N becomes large. Such objects are natural analogues 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 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 using a duality formula expressing the N by N Hankel determinant as a 2|q|-fold integral, where q is a fixed vector, which is valid when each component of q is natural.We thus verify, for such q, a recent conjecture of Forrester and Frankel derived using a log-gas argument.

16 pages. Published version, with new references added, and some minor errors corrected

References in corpus (11)

• Finite one dimensional impenetrable Bose systems: Occupation numbers
• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Universal correlations of trapped one-dimensional impenetrable bosons
• Ground-state properties of hard core bosons in one-dimensional harmonic traps
• Applications and generalizations of Fisher-Hartwig asymptotics
• Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons
• Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions
• Singularity dominated strong fluctuations for some random matrix averages
• Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges
• Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations
• Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

Cited by in corpus (6)

• Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes
• Asymptotic corrections to the eigenvalue density of the GUE and LUE
• Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
• Random Matrices with Merging Singularities and the Painlevé V Equation
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
• On the moments of characteristic polynomials