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Monotonicity of the logarithmic energy for random matrices

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorChafai, Djalil
HAL ID: 11025
ORCID: 0000-0002-1446-1428
hal.structure.identifierNew York University [Abu Dhabi]
dc.contributor.authorDadoun, Benjamin
HAL ID: 1063033
hal.structure.identifierNew York University [Abu Dhabi]
dc.contributor.authorYoussef, Pierre
dc.date.accessioned2023-01-20T10:18:36Z
dc.date.available2023-01-20T10:18:36Z
dc.date.issued2022
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/23770
dc.language.isoenen
dc.subjectRandom matricesen
dc.subjectEntropyen
dc.subjectVariational analysisen
dc.subjectLogarithmic energyen
dc.subject.ddc519en
dc.titleMonotonicity of the logarithmic energy for random matricesen
dc.typeDocument de travail / Working paper
dc.description.abstractenIt is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko--Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the mentioned models which can be of independent interest.en
dc.publisher.cityParisen
dc.identifier.citationpages23en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris Dauphine-PSLen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2022
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2023-01-20T10:14:30Z
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