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Asymptotics of a cubic sine kernel determinant. (English) Zbl 1318.82015

St. Petersbg. Math. J. 26, No. 4, 515-565 (2015) and Algebra Anal. 26, No. 4, 2-91 (2014).
Summary: 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 \( \gamma \).

MSC:

82B23 Exactly solvable models; Bethe ansatz
33E05 Elliptic functions and integrals
34E05 Asymptotic expansions of solutions to ordinary differential equations
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
82C23 Exactly solvable dynamic models in time-dependent statistical mechanics
15B52 Random matrices (algebraic aspects)
15A15 Determinants, permanents, traces, other special matrix functions
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