Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere - Bétermin, Laurent; Sandier, Etienne (Constructive approximation, 2018)
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 ‖ x ‖ 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 → + ∞ of the minimal logarithmic energy of n points on the unit sphere in 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.