论文标题
随机Weierstrass Zeta函数II。电通量通过可整流曲线的波动
The random Weierstrass zeta function II. Fluctuations of the electric flux through rectifiable curves
论文作者
论文摘要
考虑一个随机的平面点过程，其法律在平面等法下是不变的。我们认为该过程是点电荷的随机分布，并考虑电荷分布产生的电场。在这项工作的第一部分中，我们在光谱侧发现了一个条件，该条件表征何时该场本身具有不变的二阶结构。在这里，我们固定了一个具有不变场的过程，并研究了磁通量通过平面中的大弧和曲线的波动。在该过程的适当条件下，在曲线上表示$γ$，我们表明通量通过$ r \ \ r \，γ$的渐近差异像$ r $ r $ r $ r $倍的签名长度为$γ$。作为推论，我们发现扩张的约旦结构域中的电荷波动与周长是渐近的，只有边界是可纠正的。 该证明是基于密切相关数量的渐近分析（沿曲线的复杂电作用）。在分析中的决定性作用是通过经典AHLFORS规律性条件的签名版本发挥的。
Consider a random planar point process whose law is invariant under planar isometries. We think of the process as a random distribution of point charges and consider the electric field generated by the charge distribution. In Part I of this work, we found a condition on the spectral side which characterizes when the field itself is invariant with a well-defined second-order structure. Here, we fix a process with an invariant field, and study the fluctuations of the flux through large arcs and curves in the plane. Under suitable conditions on the process and on the curve, denoted $Γ$, we show that the asymptotic variance of the flux through $R\,Γ$ grows like $R$ times the signed length of $Γ$. As a corollary, we find that the charge fluctuations in a dilated Jordan domain is asymptotic with the perimeter, provided only that the boundary is rectifiable. The proof is based on the asymptotic analysis of a closely related quantity (the complex electric action of the field along a curve). A decisive role in the analysis is played by a signed version of the classical Ahlfors regularity condition.
