论文标题
二维库仑气体：通过光谱间隙波动
The two-dimensional Coulomb gas: fluctuations through a spectral gap
论文作者
论文摘要
我们研究了一类径向对称的库仑气体在反温度下$β= 2 $，为此，液滴由许多同心的annuli组成，至少有一个有界的``gap''$ g $，即，液滴的补充的连接组件，滴落了液滴。令$ n $为粒子总数。除其他事项外，我们将细小的渐近造物推断为$ n \ to \ infty $，用于边缘密度和间隙附近的相关内核，以及平滑线性统计波动的累积产生函数。我们通常会发现落在间隙边缘附近的颗粒分布中的振荡行为。这些振荡是根据离散的高斯分布，加权的szegő内核和雅各比theta函数明确给出的，该函数取决于参数$ n $。
We study a class of radially symmetric Coulomb gas ensembles at inverse temperature $β=2$, for which the droplet consists of a number of concentric annuli, having at least one bounded ``gap'' $G$, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let $n$ be the total number of particles. Among other things, we deduce fine asymptotics as $n \to \infty$ for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegő kernels, and the Jacobi theta function, which depend on the parameter $n$.
