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abstract: '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.'
author:
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    Laurent Bétermin\
    Etienne Sandier
bibliography:
- 'allbiblio.bib'
title: Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere
---

**AMS 