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

Abstract

We study the Hamiltonian of a two-dimensional log-gas with a confining\npotential $V$ satisfying the weak growth assumption -- $V$ is of the same order\nthan $2\\log|x|$ near infinity -- considered by Hardy and Kuijlaars [J. Approx.\nTheory, 170(0) : 44-58, 2013]. We prove an asymptotic expansion, as the number\n$n$ of points goes to infinity, for the minimum of this Hamiltonian using the\nGamma-Convergence method of Sandier and Serfaty [Ann. Proba., to appear, 2015].\nWe show that the asymptotic expansion as $n\\to +\\infty$ of the minimal\nlogarithmic energy of $n$ points on the unit sphere in $\\mathbb{R}^3$ has a\nterm of order $n$ thus proving a long standing conjecture of Rakhmanov, Saff\nand Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the\nequivalence between the conjecture of Brauchart, Hardin and Saff [Contemp.\nMath., 578:31-61,2012] about the value of this term and the conjecture of\nSandier and Serfaty [Comm. Math. Phys., 313(3):635-743, 2012] about the\nminimality of the triangular lattice for a "renormalized energy" $W$ among\nconfigurations of fixed asymptotic density.\n