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

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