The Random Weierstrass Zeta Function II. Fluctuations of the Electric Flux Through Rectifiable Curves

Abstract

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 $$\Gamma $$ Γ , we show that the asymptotic variance of the flux through $$R\,\Gamma $$ R Γ grows like R times the signed length of $$\Gamma $$ 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.