Large gap asymptotics for the generating function of the sine point process

We consider the generating function of the sine point process on [Formula: see text] consecutive intervals. It can be written as a Fredholm determinant with discontinuities, or equivalently as the convergent series [Formula: see text] where [Formula: see text]. In particular, we can deduce from it joint probabilities of the counting function of the process. In this work, we obtain large gap asymptotics for the generating function, which are asymptotics as the size of the intervals grows. Our results are valid for an arbitrary integer [Formula: see text] , in the cases where all the parameters [Formula: see text] , except possibly one, are positive. This generalizes two known results: (1) a result of Basor and Widom, which corresponds to [Formula: see text] and [Formula: see text] , and (2) the case [Formula: see text] and [Formula: see text] for which many authors have contributed. We also present some applications in the context of thinning and conditioning of the sine process.