Duke Mathematical Journal
- Duke Math. J.
- Volume 164, Number 15 (2015), 2897-2987.
Toeplitz determinants with merging singularities
We study asymptotic behavior for the determinants of Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance from each other on the unit circle. We obtain large asymptotics which are uniform for , where is fixed. They describe the transition as between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.
Duke Math. J., Volume 164, Number 15 (2015), 2897-2987.
Received: 28 April 2014
Revised: 23 October 2014
First available in Project Euclid: 1 December 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 15B05: Toeplitz, Cauchy, and related matrices 33E17: Painlevé-type functions
Secondary: 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20] 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
Claeys, T.; Krasovsky, I. Toeplitz determinants with merging singularities. Duke Math. J. 164 (2015), no. 15, 2897--2987. doi:10.1215/00127094-3164897. https://projecteuclid.org/euclid.dmj/1448980436