The sine process under the influence of a varying potential

Abstract

We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma ):=\det (I-\gamma K_s{\upharpoonright }_{L^2(-1,1)})$, $0\le \gamma \le 1$. 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 \cite{Dyson2} of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned GUE \cite{BP}.

arXiv comment7 pages, 1 figure, a contribution to JMP's volume devoted to the memory of LD Faddeev

References in corpus (8)

• Determinantal random point fields
• Randomly incomplete spectra and intermediate statistics
• Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics
• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Deformations of the Tracy-Widom distribution
• On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I
• From gap probabilities in random matrix theory to eigenvalue expansions
• Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

Cited by in corpus (3)

• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Large gap asymptotics for the generating function of the sine point process
• Band-limited mimicry of point processes by point processes supported on a lattice