The random Weierstrass zeta function I. Existence, uniqueness, fluctuations

TL;DR

This paper constructs and characterizes random meromorphic functions with prescribed poles based on stationary point processes, serving as stochastic analogues of the Weierstrass zeta function and modeling electric fields from random charge distributions.

Contribution

It introduces a novel construction of random meromorphic functions with prescribed poles and characterizes stationary processes where the associated functions are stationary after mean subtraction.

Findings

Identifies conditions for stationarity of the constructed functions

Provides a framework linking point processes to random elliptic functions

Models electric fields generated by random charge distributions

Abstract

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.