Toeplitz determinants with merging singularities
(2015) Duke Mathematical Journal — Vol. 164, n° 15, p. 2897-2987 (2015)
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- Abstract
- We study asymptotic behavior for determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance 2t≥0 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 t→0 between the asymptotic regimes of 2 singularities and 1 singularity. The asymptotics involve a particular solution to the Painlev\'e 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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Claeys, T., Krasovsky, I., & et al. (2015). Toeplitz determinants with merging singularities. Duke Mathematical Journal, 164(15), 2897-2987. https://doi.org/10.1215/00127094-3164897 (Original work published 2015)