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

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.