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@article{Claeys:2015:10.1215/00127094-3164897,
author = {Claeys, T and Krasovsky, I},
doi = {10.1215/00127094-3164897},
journal = {Duke Mathematical Journal},
pages = {2897--2987},
title = {Toeplitz determinants with merging singularities},
url = {http://dx.doi.org/10.1215/00127094-3164897},
volume = {164},
year = {2015}
}

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TY  - JOUR
AB  - We study asymptotic behavior for the 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 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.
AU  - Claeys,T
AU  - Krasovsky,I
DO  - 10.1215/00127094-3164897
EP  - 2987
PY  - 2015///
SN  - 0012-7094
SP  - 2897
TI  - Toeplitz determinants with merging singularities
T2  - Duke Mathematical Journal
UR  - http://dx.doi.org/10.1215/00127094-3164897
UR  - http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000366144600002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
UR  - https://projecteuclid.org/euclid.dmj/1448980436
VL  - 164
ER  -

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