Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

Abstract

We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble (GUE) statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

Simple derivation of the (– λ H)5/2 tail for the 1D KPZ equation

“…We recall the result of Refs. [30][31][32][33] that for a set of points {a i } i∈N following a determinantal point process with kernel K, we have in our system of unit…”

“…The formula for the general cumulant can be found in Ref. [31] where similar problems of linear statistics have been studied…”

Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions

“…There has been a lot of interest in linear statistics and central limit theorems for eigenvalues of large random matrices in the Gaussian Unitary Ensemble (GUE) [40], see e.g. [41][42][43] and references therein.…”

Linear statistics and pushed Coulomb gas at the edge of β -random matrices: Four paths to large deviations

“…conductance, shot noise, Renyi entropies, interfaces center of mass, particule number fluctuations. At large N , central limit theorems, universality, and connections to the Gaussian free field were shown [3,22,[32][33][34][35][36][37][38][39] for typical fluctuations of L in the bulk of the spectrum. Large deviations were also studied in the bulk [40][41][42][43][44], from the Coulomb gas representation, and recently for truncated linear statistics [45,46], showing interesting phase transitions.…”