Communications in Mathematical Physics

, Volume 291, Issue 3, pp 691-761

First online:

Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications

  • N. KitanineAffiliated withLPTM, Université de Cergy-Pontoise et CNRS
  • , K. K. KozlowskiAffiliated withLaboratoire de Physique, ENS Lyon et CNRS
  • , J. M. MailletAffiliated withLaboratoire de Physique, ENS Lyon et CNRS Email author 
  • , N. A. SlavnovAffiliated withSteklov Mathematical Institute
  • , V. TerrasAffiliated withLaboratoire de Physique, ENS Lyon et CNRS

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We investigate the asymptotic behaviour of a generalised sine kernel acting on a finite size interval [−q ; q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener–Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.