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

Abstract

We consider the generating function of the sine point process on m consecutive intervals. It can be written as a Fredholm determinant with discontinuities, or equivalently as the convergent series ∑k1,…,km⩾0P∩j=1m#{pointsinthejthinterval}=kj∏j=1msjkj,where s1,…,sm∈false[0,+∞false). 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 m, in the cases where all the parameters s1,…,sm, except possibly one, are positive. This generalizes two known results: (1) a result of Basor and Widom, which corresponds to m=1 and s1>0, and (2) the case m=1 and s1=0 for which many authors have contributed. We also present some applications in the context of thinning and conditioning of the sine process.

Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

“…The function F (x; s) for s ∈ (0, 1) is the probability distribution of the largest particle in the thinned Airy point process, which is obtained by removing each particle independently with probability s [13]. For m ≥ 1, F ( x; s) is the probability to observe a gap on (x m , +∞) in the piecewise constant thinned Airy point process, where each particle on (x j , x j−1 ) is removed with probability s j (see [16] for a similar situation, with more details provided). It was shown recently that the m-point determinants F ( x; s) for m > 1 can be expressed identically in terms of solutions to systems of coupled Painlevé II equations [17,42], which are special cases of integro-differential generalizations of the Painlevé II equations which are connected to the KPZ equation [2,18].…”

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

“…, m} \ {p − 1, p}, and where µ j , σ 2 j and Σ j,k are given by (1.20)- (1.22). Note that for the (unthinned) sine point process, the covariance (1.32) has a leading term proportional to log r, while in the above, the covariance is of order 1 (it is also worth to compare with similar covariances in the Airy and Bessel point processes, see [13,11]).…”

“…This construction is standard (see e.g. [32,28]) and similar to those done in [13,11]. It involves a model RH problem Φ HG which we can be found in the appendix, Section 8.2.…”

Full counting statistics for interacting trapped fermions

“…To discuss the FCS within an interval [a, b] at large N , i.e., the fluctuations of N [a,b] , one needs to distinguish two natural length scales: the microscopic scale given by the interparticle distance ∼ 1/ N , and the macroscopic scale of order x + e − x − e ∼ N . It is well known since Dyson and Mehta [35][36][37][38] that for an interval of microscopic size the variance for the GUE is given by [6,[39][40][41][42][43][44][45][46] Var…”

“…One can also ask about higher cumulants of N [a,b] , i.e., beyond the variance given in (2), both for microscopic and macroscopic interval [a, b]. In the absence of potential, i.e., for free fermions, there exist results for the higher cumulants which are obtained using the sinekernel [6,42,46]. A natural conjecture, which we put forward in [52] is that these higher cumulants are determined solely from fluctuations on microscopic scales.…”