Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2015

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Résumé

We study the Hamiltonian of a two-dimensional log-gas with a confining potential $V$ satisfying the weak growth assumption -- $V$ is of the same order than $2\log|x|$ near infinity -- considered by Hardy and Kuijlaars [J. Approx. Theory, 170(0) : 44-58, 2013]. We prove an asymptotic expansion, as the number $n$ of points goes to infinity, for the minimum of this Hamiltonian using the Gamma-Convergence method of Sandier and Serfaty [Ann. Proba., to appear, 2015]. We show that the asymptotic expansion as $n\to +\infty$ of the minimal logarithmic energy of $n$ points on the unit sphere in $\mathbb{R}^3$ has a term of order $n$ thus proving a long standing conjecture of Rakhmanov, Saff and Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff [Contemp. Math., 578:31-61,2012] about the value of this term and the conjecture of Sandier and Serfaty [Comm. Math. Phys., 313(3):635-743, 2012] about the minimality of the triangular lattice for a ``renormalized energy" $W$ among configurations of fixed asymptotic density.
Fichier principal
Vignette du fichier
Sphere Betermin Sandier.pdf (271.67 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00979926 , version 1 (17-04-2014)
hal-00979926 , version 2 (21-04-2014)
hal-00979926 , version 3 (19-01-2015)
hal-00979926 , version 4 (28-10-2015)
hal-00979926 , version 5 (17-04-2018)

Identifiants

Citer

Laurent Bétermin, Etienne Sandier. Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere. 2015. ⟨hal-00979926v5⟩
179 Consultations
377 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More