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The sine process under the influence of a varying potential. (English) Zbl 1404.60010

Summary: We review the authors’ recent work where we obtain the uniform large \(s\) asymptotics for the Fredholm determinant \(D(s,\gamma):= \det(I - \gamma K_s\upharpoonright_{L^2(- 1,1)}),\; 0 \leq \gamma\leq 1.\) The operator \(K_{s}\) acts with kernel \(K_{s}(x, y) = sin(s(x - y))/(\pi(x - y))\), and \(D(s,\gamma)\) appears for instance in Dyson’s model of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble.{
©2018 American Institute of Physics}

MSC:

60B20 Random matrices (probabilistic aspects)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
15B52 Random matrices (algebraic aspects)
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