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Asymptotics of a cubic sine kernel determinant


Authors: T. Bothner and A. Its
Original publication: Algebra i Analiz, tom 26 (2014), nomer 4.
Journal: St. Petersburg Math. J. 26 (2015), 515-565
MSC (2010): Primary 82B23; Secondary 33E05, 34E05, 34M50
DOI: https://doi.org/10.1090/spmj/1350
Published electronically: May 6, 2015
MathSciNet review: 3289185
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Abstract: The one-parameter family of Fredholm determinants $ \det (I-\gamma K_{\mathrm {csin}})$, $ \gamma \in \mathbb{R}$, is studied for an integrable Fredholm operator $ K_{\mathrm {csin}}$ that acts on the interval $ (-s,s)$ and whose kernel is a cubic generalization of the sine kernel that appears in random matrix theory. This Fredholm determinant arises in the description of the Fermi distribution of semiclassical nonequilibrium Fermi states in condensed matter physics as well as in the random matrix theory. By using the Riemann-Hilbert method, the large $ s$ asymptotics of $ \det (I-\gamma K_{\mathrm {csin}} )$ is calculated for all values of the real parameter $ \gamma $.


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Additional Information

T. Bothner
Affiliation: Centre de recherchrs mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal, Québec H3T 1J4, Canada
Email: bothner@crm.umontreal.ca

A. Its
Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, Indiana 46202
Email: itsa@math.iupui.edu

DOI: https://doi.org/10.1090/spmj/1350
Keywords: Fredholm determinant, integrable Fredholm operator, Riemann--Hilbert method, Fermi distribution
Received by editor(s): July 10, 2013
Published electronically: May 6, 2015
Additional Notes: This work was supported in part by the National Science Foundation (NSF) Grant DMS-1001777 and by the SPbGU grant N11.38.215.2014
Dedicated: To the memory of Vladimir Savelievich Buslaev
Article copyright: © Copyright 2015 American Mathematical Society