Jordan曲线上外部势中Coulomb气体的极限定理
Limit theorems for Coulomb gases on a Jordan curve in an external potential
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中文总结 AI 辅助
本文研究Jordan曲线上外部势中Coulomb气体的自由能渐近展开与线性统计量中心极限定理，提出基于广义Grunsky算子的方法，证明方差与势无关。
中文摘要 AI 辅助
我们考虑在外部势$V$中，Jordan曲线$\gamma$上的Coulomb气体，其逆温度$\beta>0$，并获得了自由能直到$o(1)$的渐近展开以及线性统计量的中心极限定理。我们关注单割区域，其中$\gamma$在$V$中的加权平衡测度的密度在$\gamma$上严格为正。（归一化）展开中的常数项由两部分组成：广义Grunsky算子的Fredholm行列式，以及$\gamma$的加权平衡测度密度的对数的Dirichlet能量。后者的系数在$\beta=2$时为零。线性统计量波动的方差仅依赖于测试函数的Dirichlet能量，因此与$V$无关。我们方法的关键在于，广义Grunsky算子和伴随的平衡参数化使我们能够将外部势中曲线上的粒子以保持平衡测度的方式传输到参考对象。在我们的设置中，单位圆是自然的参考对象。
英文摘要
We consider a Coulomb gas on a Jordan curve $γ$ in an external potential $V$ at inverse temperature $β>0$ and obtain an asymptotic expansion of the free energy up to $o(1)$ and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of $γ$ in $V$ is strictly positive on $γ$. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of $γ$. The coefficient of the latter vanishes for $β=2$. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of $V$. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.
发表机构
- KTH Royal Institute of Technology（瑞典皇家理工学院）
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