Asymptotics of a cubic sine kernel determinant

Bothner, T.; Its, A.

doi:10.1090/spmj/1350

Asymptotics of a cubic sine kernel determinant
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by T. Bothner and A. Its

St. Petersburg Math. J. 26 (2015), 515-565

DOI: https://doi.org/10.1090/spmj/1350

Published electronically: May 6, 2015

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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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