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.

Appendix

Here we prove the following:

Proposition 7.8

Assume X is a Polish space X, on which $\mathbb {R}^n$ acts continuously. We denote this action by $(\lambda ,u)\rightarrow \theta _\lambda u$ and assume it is separately continuous w.r.t both $\lambda \in \mathbb {R}^n$ and $u\in X$. Assume P is a probability measure on $\mathbb {R}^n\times X$ which for every $\lambda$ is invariant under the map $(x,u)\rightarrow (x,\theta _\lambda u)$. Then, for any continuous function $x\rightarrow \lambda (x)$, it holds that P is invariant under the map $(x,u)\rightarrow (x,\theta _\lambda (x) u)$.

Proof

Let $\Phi$ be any bounded continuous function on $\mathbb {R}^n\times X$. We need to prove that for any continuous function $x\rightarrow \lambda (x)$,

$$\begin{aligned} \int \Phi (x,u)\,{\mathrm{d}}P(x,u) = \int \Phi (x,\theta _{\lambda (x)}u)\,{\mathrm{d}}P(x,u). \end{aligned}$$

For any integer $k>0$, we let $\{\chi _{i,k}\}_i$ be a partition of unity on $\mathbb {R}^n$ subordinate to the covering of $\mathbb {R}^n$ by balls of radius 1 / k, and we let $x_{i,k}$ belong to the support of $\chi _{i,k}$. Then, from the continuity of $\Phi$, $\lambda$, and $\theta$, it is straightforward to check that for every $(x,u)\in \mathbb {R}^n\times X$, we have

$$\begin{aligned} \lim _{k\rightarrow +\infty } \sum _i \chi _{i,k}(x) \Phi (x,\theta _{\lambda (x_{i,k})}u) = \Phi (x,\theta _{\lambda (x)}u). \end{aligned}$$

It follows by dominated convergence that

$$\begin{aligned} \lim _{k\rightarrow +\infty } \sum _i\int \chi _{i,k}(x) \Phi (x,\theta _{\lambda (x_{i,k})}u) \,{\mathrm{d}}P(x,u) = \int \Phi (x,\theta _{\lambda (x)}u)\,{\mathrm{d}}P(x,u). \end{aligned}$$

(7.4)

But by the invariance of P, we have

$$\begin{aligned} \int \chi _{i,k}(x) \Phi (x,\theta _{\lambda (x_{i,k})}u) \,{\mathrm{d}}P(x,u) = \int \chi _{i,k}(x) \Phi (x,u) \,{\mathrm{d}}P(x,u), \end{aligned}$$

hence

$$\begin{aligned} \sum _i\int \chi _{i,k}(x) \Phi (x,\theta _{\lambda (x_{i,k})}u) \,{\mathrm{d}}P(x,u) = \int \Phi (x,u) \,{\mathrm{d}}P(x,u). \end{aligned}$$

Replacing (7.4), we get the desired result. $\square$