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Algebra i Analiz, 2014, Volume 26, Issue 4, Pages 22–91 (Mi aa1391)  

This article is cited in 6 scientific papers (total in 6 papers)

Research Papers

Asymptotics of a cubic sine kernel determinant

T. Bothner, A. Its

Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN, 46202 USA

Abstract: The one-parameter family of Fredholm determinants $\operatorname{det}(I-\gamma K_\mathrm{csin})$, $\gamma\in\mathbb R$, 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 $\operatorname{det}(I-\gamma K_\mathrm{csin})$ is calculated for all values of the real parameter $\gamma$.

Keywords: Fredholm determinant, integrable Fredholm operator, Riemann–Hilbert method, Fermi distribution.

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English version:
St. Petersburg Mathematical Journal, 2015, 26:4, 515–565

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Received: 10.07.2013
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Citation: T. Bothner, A. Its, “Asymptotics of a cubic sine kernel determinant”, Algebra i Analiz, 26:4 (2014), 22–91; St. Petersburg Math. J., 26:4 (2015), 515–565

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. Bothner, “From gap probabilities in random matrix theory to eigenvalue expansions”, J. Phys. A-Math. Theor., 49:7 (2016), 075204  crossref  mathscinet  zmath  isi  elib  scopus
    2. T. Bothner, “Transition asymptotics for the Painlevé II transcendent”, Duke Math. J., 166:2 (2017), 205–324  crossref  mathscinet  zmath  isi  scopus
    3. T. Bothner, P. Deift, A. Its, I. Krasovsky, “On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II”, Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics: the Albrecht Bottcher Anniversary Volume, Operator Theory Advances and Applications, 259, eds. D. Bini, T. Ehrhardt, A. Karlovich, I. Spitkovsky, Springer International Publishing Ag, 2017, 213–234  crossref  mathscinet  zmath  isi
    4. T. Bothner, R. Buckingham, “Large deformations of the Tracy-Widom distribution I: Non-oscillatory asymptotics”, Commun. Math. Phys., 359:1 (2018), 223–263  crossref  mathscinet  zmath  isi  scopus
    5. A. R. Its, O. Lisovyy, A. Prokhorov, “Monodromy dependence and connection formulae for isomonodromic tau functions”, Duke Math. J., 167:7 (2018), 1347–1432  crossref  mathscinet  zmath  isi  scopus
    6. Bothner T. Its A. Prokhorov A., “On the Analysis of Incomplete Spectra in Random Matrix Theory Through An Extension of the Jimbo-Miwa-Ueno Differential”, Adv. Math., 345 (2019), 483–551  crossref  mathscinet  zmath  isi  scopus
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