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Consider a random planar point process whose law is invariant under planarisometries. We think of the process as a random distribution of point chargesand consider the electric field generated by the charge distribution. In Part Iof this work, we found a condition on the spectral side which characterizeswhen 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 ofthe flux through large arcs and curves in the plane. Under suitable conditionson the process and on the curve, denoted $\Gamma$, we show that the asymptoticvariance of the flux through $R\,\Gamma$ grows like $R$ times the signed lengthof $\Gamma$. As a corollary, we find that the charge fluctuations in a dilatedJordan domain is asymptotic with the perimeter, provided only that the boundaryis 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 inthe analysis is played by a signed version of the classical Ahlfors regularitycondition.

The random Weierstrass zeta function II. Fluctuations of the electric flux through rectifiable curves