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

• Published in: Constructive approximation Vol. 47; no. 1; pp. 39 - 74
• Main Authors: Bétermin, Laurent, Sandier, Etienne
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2018; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Asymptotic properties; Asymptotic series; Confining; Infinity; Mathematics; Mathematics and Statistics; Numerical Analysis; Quantum theory; Weak confinement; Primary 52A40; 82B05; Renormalized energy; 31C20; Crystallization; 82B21; Ginzburg–Landau; Triangular lattice; Secondary 41A60; Logarithmic energy; Vortices; Abrikosov lattices; Logarithmic potential theory; Coulomb gas; Gamma-convergence; Number theory
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-016-9357-z

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.