1 December 2015 Toeplitz determinants with merging singularities
T. Claeys, I. Krasovsky
Duke Math. J. 164(15): 2897-2987 (1 December 2015). DOI: 10.1215/00127094-3164897


We study asymptotic behavior for the determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance 2t0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0<t<t0, where t0 is fixed. They describe the transition as t0 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.


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T. Claeys. I. Krasovsky. "Toeplitz determinants with merging singularities." Duke Math. J. 164 (15) 2897 - 2987, 1 December 2015. https://doi.org/10.1215/00127094-3164897


Received: 28 April 2014; Revised: 23 October 2014; Published: 1 December 2015
First available in Project Euclid: 1 December 2015

zbMATH: 1333.15018
MathSciNet: MR3430454
Digital Object Identifier: 10.1215/00127094-3164897

Primary: 15B05 , 33E17
Secondary: 35Q15 , 42C05

Keywords: double scaling , Fisher–Hartwig singularities , one-dimensional Bose gas , Painlevé functions , random matrices , Riemann–Hilbert problems , Toeplitz determinants

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 15 • 1 December 2015
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