Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

Abstract

We study the moment generating function of the disk counting statistics of a two-dimensional determinantal point process which generalizes the complex Ginibre point process. This moment generating function involves an n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has discontinuities along circles centered at 0. These discontinuities can be thought of as a two-dimensional analogue of jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for this determinant, up to and including the term of order n − 1 2 . We allow for any number of discontinuities in the bulk, one discontinuity at the edge, and any number of discontinuities bounded away from the bulk. As an application, we obtain the large n asymptotics of all the cumulants of the disk counting function up to and including the term of order n − 1 2 , both in the bulk and at the edge. This improves on the best known results for the complex Ginibre point process, and for general values of our parameters these results are completely new. Our proof makes a novel use of the uniform asymptotics of the incomplete gamma function.

Counting statistics for non-interacting fermions in a rotating trap

“…In the LLL limit Ω → 1, we find that our result is in agreement with [27], the connection between the scaling functions being f edge,Var 0 (s) = K e 2 (s) as defined in [27] (see also [32]), note however [45]. Here we obtain the shape of the steps for all plateaus k ≥ 0, using a rather different method.…”

“…Finally, from the above predicted form of the cumulants, together with the relation (32), we conclude that the entanglement entropy should also be a piecewise linear function of R in this regime. More precisely…”

“…In that case, the full counting statistics (FCS) could be obtained exactly. In particular it was found [27] that all cumulants of the number N R of fermions inside a disk of radius R (centered on the origin r = 0) grows as ∼ R for large R. This was recently proved rigorously in [32] where higher order asymptotics were obtained. This is in marked contrast with the behavior of e.g free fermions in 2d (with Ω = 0), where the variance of N R grows as R log R, a result which was extended and refined in the presence of an arbitrary external potential V (r) in our recent work [33].…”

“…There are several avenues left for future research. A natural next step would be to calculate the coefficients C k,2m in (56) associated to the piecewise linear dependence of the higher cumulants in the regime 1 − Ω/ω = O(1/N ), and therefore, by virtue of (32), also obtain the exact coefficients γ q,k for the entanglement entropy, thus extending the results for γ q,1 for the lowest Landau level in [27]. It would also be interesting to study the full distribution of N R , and in particular its large-deviation form, as was done in [49] for the Ginibre ensemble (corresponding to Ω → ω in our system), see also [50,51] in the context of Ginibre matrices.…”

Large gap asymptotics on annuli in the random normal matrix model

“…The random normal matrix model has been extensively studied over the years, see e.g. [18,31] for early works, [37,63,4,53] for smooth linear statistics, [62,6,71,27,35,15] for non-smooth linear statistics and their moment generating functions, and [8,44,54,55,3] for recent investigations on planar orthogonal polynomials. Despite all these progresses, the problem of determining large gap asymptotics in this model has remained an outstanding problem.…”