重整化能量与球面上最优对数能量的渐近展开
Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere
摘要：
我们研究了二维对数气体的哈密顿量，其限制势$V$满足弱增长假设-- $V$与$2\log|x|$在无穷远处具有相同的阶数--由Hardy和Kuijlaars [J. Approx. Theory，170（0）：44-58，2013]。我们证明了一个渐近展开，作为点的数量$n$走向无穷大，为最小的这个哈密顿量使用伽玛收敛方法的Sandier和Serfaty [Ann. Proba.，出现，2015年）。证明了单位球面上$n$点的最小对数能量在$\mathbb{R}^3$中的渐近展开式为$n\to +\infty$，具有一个$n$阶项，从而证明了Rakhmanov，Saff和Zhou的一个长期存在的猜想[Math. Res. Letters，1：647-662，1994].最后证明了Brauchart，Hardin和Saff [Contemp.数学，578：31- 61，2012]关于这个项的值以及Sandier和Serfaty的猜想[Comm. Math. Phys.，313（3）：635-743，2012]关于在固定渐近密度的配置中的“重整化能量”$W$的三角晶格的最小性。
英文摘要：
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 than $2\log|x|$ near infinity -- considered by Hardy and Kuijlaars [J. Approx. Theory, 170(0) : 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. Proba., to appear, 2015]. We show that the asymptotic expansion as $n\to +\infty$ 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, Saff and Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff [Contemp. Math., 578:31-61,2012] about the value of this term and the conjecture of Sandier and Serfaty [Comm. 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.
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