On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential. I.(English)Zbl 1321.82027

The authors study the determinant of the integral Fredholm’s operator with kernel $$\sin x(a-b) / \pi(a-b)$$ which appears in the analysis of a log-gas of interacting particles. The main result is obtained in the form of a general expression for the logarithm of this determinant. The proof is lengthy and works via Riemann-Hilbert analysis.

MSC:

 82C22 Interacting particle systems in time-dependent statistical mechanics 15B52 Random matrices (algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 15A15 Determinants, permanents, traces, other special matrix functions 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

DLMF
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