Journal article
The classical beta-ensembles with beta proportional to 1/N: From loop equations to Dyson's disordered chain
PJ Forrester, G Mazzuca
JOURNAL OF MATHEMATICAL PHYSICS | AIP Publishing | Published : 2021
DOI: 10.1063/5.0048481
Abstract
In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading 1/N correction to the density. From earlier literature, the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying, in particular, the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterization ..
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Awarded by Australian Research Council
Awarded by European Union's H2020 research and innovation program under Marie Sklowdoska-Curie Grant
Funding Acknowledgements
The research of P.J.F. is part of the program of study supported by the Australian Research Council Centre of Excellence ACEMS and Discovery Project Grant No. DP210102887. The research of G.M. is part of the program of study supported by the European Union's H2020 research and innovation program under Marie Sklowdoska-Curie Grant No. 778010 IPaDEGAN. We thank G. Akemann for (indirectly) facilitating this collaboration by inviting G.M. to speak as part of the Bielefeld-Melbourne random matrix seminar in December 2020. We thank K. D. Trinh for alerting us that a recursive formula for the covariances {mu<SUP>G</SUP><INF>( p,q),0</INF>} was first given in Ref. 45. We also thank the anonymous referee for his valuable comments, which led to improvements of this paper.