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