  We study the diffusion of massive particles in the space time of an Abelian
Higgs string. The particles in the early universe plasma execute Brownian
motion. This motion of the particles is modeled as a two dimensional random
walk in the plane of the Abelian Higgs string. The particles move randomly in
the space time of the string according to their geodesic equations. We observe
that for certain values of their energy and angular momentum, an overdensity of
particles is observed close to the string. We find that the string parameters
determine the distribution of the particles. We make an estimate of the density
fluctuation generated around the string as a function of the deficit angle.
Though the thickness of the string is small, the length is large and the
overdensity close to the string may have cosmological consequences in the early
universe.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics