TY - JOUR

T1 - The Random Weierstrass Zeta Function II. Fluctuations of the Electric Flux Through Rectifiable Curves

AU - Sodin, Mikhail

AU - Wennman, Aron

AU - Yakir, Oren

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/10

Y1 - 2023/10

N2 - 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.

AB - 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.

KW - Charge fluctuations

KW - Electric field

KW - Hyperuniformity

KW - Number variance

KW - Stationary point process

UR - http://www.scopus.com/inward/record.url?scp=85174549359&partnerID=8YFLogxK

U2 - 10.1007/s10955-023-03170-y

DO - 10.1007/s10955-023-03170-y

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AN - SCOPUS:85174549359

SN - 0022-4715

VL - 190

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 10

M1 - 164

ER -