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

Mikhail Sodin, Aron Wennman*, Oren Yakir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 RΓ 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.

Original languageEnglish
Article number164
JournalJournal of Statistical Physics
Issue number10
StatePublished - Oct 2023


FundersFunder number
European Research Council2022-03611, 692616
United States-Israel Binational Science Foundation202019
Israel Science Foundation1288/21
Knut och Alice Wallenbergs Stiftelse2017.0398
Tel Aviv University


    • Charge fluctuations
    • Electric field
    • Hyperuniformity
    • Number variance
    • Stationary point process


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