Large gap asymptotics on annuli in the random normal matrix model

Abstract

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the formwhere n is the number of points of the process. We determine the constants C1, . . . , C6 explicitly, as well as the oscillatory term Fn which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: when the hole region is a disk, only the constants C1, . . . , C4 were previously known, and when the hole region is an annulus, only C1 was previously known. For general values of our parameters, even C1 is new. A main discovery of this work is that Fn is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

Counting statistics for non-interacting fermions in a rotating trap

“…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.…”

“…with C k,2 = C k given in Eqs. ( 50) and (51). They are discussed in Appendix F. Although we have not studied these coefficients in detail, we show, in section V that their large k asymptotics match the 1 − Ω = O(1) result ( 21) in the limit 1 − Ω 1, as we also show explicitly in the case of the variance.…”

“…In this regime the mean fermion den-sity exhibits a staircase form, with discrete plateaus at values ρ = k π corresponding to filling k successive Landau levels, as found in [27,28]. Here we compute the variance Var N R , which is found to be a discontinuous piecewise linear function √ 2C k R as given in (49), where R = O(R e ) = O( √ N ), and the coefficients C k associated to the k-th Landau level are given in (51). The result for C 1 agrees with [27] using a different method.…”

Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices

“…The theoretical form of the asymptotic expansion was proved in [4,6] generalizing similar results available for the one-cut case [5]. The first leading coefficients have been obtained in the case of two intervals [8,7]. Oscillatory terms in the case of the Airy kernel have also been computed in [2].…”

Random normal matrices in the almost-circular regime

“…This is indeed a special case of a more general Mittag-Leffler potentials, see e.g. [11,16,19] and references therein. The Mittag-Leffler ensembles are contained in the class of models under consideration in this note.…”