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Edge statistics for a class of repulsive particle systems

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Kriecherbauer, Thomas [University of Bayreuth] Venker, Martin [UCL]

We study a class of interacting particle systems on (Formula presented.) which was recently investigated by Götze and Venker (Ann Probab 42(6):2207–2242, 2014. doi:10.1214/13-AOP844). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Götze and Venker (Ann Probab 42(6):2207–2242, 2014. doi:10.1214/13-AOP844) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi:10.3842/SIGMA.2016.093; Schüler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schüler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2018
Language Anglais
Journal information "Probability Theory and Related Fields" - Vol. 170, no.3-4, p. 617-655 (2017)
Publisher Springer Nature
issn 0178-8051
e-issn 1432-2064
Publication status Accepté/Sous presse
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
Keywords Statistics ; Probability and Uncertainty ; Statistics and Probability ; Analysis
Links
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Bibliographic reference Kriecherbauer, Thomas ; Venker, Martin. Edge statistics for a class of repulsive particle systems. In: Probability Theory and Related Fields, Vol. 170, no.3-4, p. 617-655 (2017)
Permanent URL http://hdl.handle.net/2078.1/197248