Cosmic strings as random walks

We present a random-walk model for the formation of cosmic strings in the early Universe. Analytic results are given for the scaling of string length with straight-line end-to-end separation, the length distribution of closed loops, and the fraction of total string length in infinite strings. We explain why the string network has the statistical properties of a set of Brownian, rather than self-avoiding, random walks, even in models in which string intersections in the initial configuration are impossible. It is found that the number of closed loops per unit volume with length between l and l+dl is dn=C${\xi }^{\mathrm{-}3/2}$ $l^{\mathrm{-}5/2}$dl, while the scaling of the straight-line distance R between two points on a string separated by a length l is R∼(lξ${)}^{1/2}$. The fraction of string length in infinite strings is ≳(2/3)–(3/4). These results are in reasonable agreement with previous Monte Carlo simulations and are confirmed by our own computer simulations in which string intersections in the initial configuration do not occur. The prevalence of infinite strings is a natural feature in the random-walk model.