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 serieswhere s1, . . . , sm ∈ [0, 1]. 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.

Counting statistics for non-interacting fermions in a $d$-dimensional potential

“…Second, since (23) coincides with the results from the sine-kernel (and the CUE) [6,19,20], it is natural to conjecture that these higher cumulants arise solely from fluctuations on microscopic scales and that (23) actually holds for any smooth potential V (x) [94]. For d > 1, using c (µ, R) Rk F (R) in Eq.…”

“…where N p [0,R] c are the cumulants of the number of fermions in the interval [0, R] for the 1d potential V (r) in (19) with m fermions. In the large µ limit, the sum in (20) is dominated by large values of and m , and is effectively cut-off at c (µ, R) Rk F (R) ≤ max , where k F (r) = 2(µ − V (r)). This allows us to use our results in 1d and to obtain the variance of N R for a general central potential, see [57].…”

“…Using our Eq. (20) we obtain [57] the cumulants of N R for free fermions in dimension d > 1, with k F R 1…”

“…An important concept to study quantum noise and correlations in many body fermionic systems is the counting statistics (CS), which characterises the fluctuations of the number of particles N D inside a domain D. Applications include shot noise [1], quantum transport [2,3], quantum dots [4,5], spin and fermionic chains [6][7][8][9], trapped fermions [10,11]. In the related context of random matrix theory (RMT), the statistics of the number of eigenvalues in an interval also generated a lot of interest [6,[12][13][14][15][16][17][18][19][20][21][22][23]. The CS is particularly important for non interacting fermions because of its connection [24][25][26][27] to the bipartite entanglement entropy (EE) of the subsystem D with its complement D. The EE is a highly non local quantity, much studied in the context of quantum information [28], conformal field theory [29][30][31], topological phases [32], quantum phase transitions [33,34], or quantum spin chains [35,36].…”

“…There are two natural length scales, the microscopic one of order the inter-particle distance ∼ 1/k F (x), and the macroscopic one of order x + − x − . For an interval D = [a, b] of microscopic size, it is well known from classical results of RMT [51,52] that for √ N |b−a| = O(1) 1 the variance behaves as [6,[12][13][14][19][20][21][22] Var…”