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 potential V satisfying the weak growth assumption—V is of the same order as $$2\log \Vert x\Vert $$ near infinity—considered by Hardy and Kuijlaars [J Approx Theory 170:44–58, 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 Probab 43(4):2026–2083, 2015]. We show that the asymptotic expansion as $$n\rightarrow +\infty $$ 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 et al. [Math Res Lett 1:647–662, 1994]. Finally, we prove the equivalence between the conjecture of Brauchart Brauchart, Hardin and Saff [Contemp. Math., 578:31–61, 2012] about the value of this term and the conjecture of Sandier and Serfaty [Commun 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.