  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.
