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GAUSSIAN AND NON-GAUSSIAN FLUCTUATIONS FOR MESOSCOPIC LINEAR STATISTICS IN DETERMINANTAL PROCESSES
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland..
2018 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, no 3, p. 1201-1278Article in journal (Refereed) Published
Abstract [en]

We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding 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 correlation kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we obtain 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 observe 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 sinekernel 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.

Place, publisher, year, edition, pages
INST MATHEMATICAL STATISTICS , 2018. Vol. 46, no 3, p. 1201-1278
Keywords [en]
Gaussian unitary ensemble, determinantal point processes, central limit theorem, cumulant method, transition
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-227746DOI: 10.1214/17-AOP1178ISI: 000430923200001Scopus ID: 2-s2.0-85045344759OAI: oai:DiVA.org:kth-227746DiVA, id: diva2:1205736
Funder
Knut and Alice Wallenberg Foundation, KAW 2010.0063
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-05-15Bibliographically approved

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