Details

Autor(en) / Beteiligte

• Claeys, T.
• Krasovsky, I.

Titel

Toeplitz determinants with merging singularities

Ist Teil von

• Duke mathematical journal, 2015-12, Vol.164 (15), p.2897-2987

Ort / Verlag

Duke University Press

Erscheinungsjahr

2015

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We study asymptotic behavior for the determinants of n\times n Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance 2t\ge0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0\lt t\lt t_{0} , where t_{0} is fixed. They describe the transition as t\to0 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.