At early stages the dynamics of cosmic string networks is expected to be
influenced by an excessive production of small loops at the scales of initial
conditions l_{min}. To understand the late time behavior we propose a very
simple analytical model of strings with a non-scaling population of loops. The
complicated non-linear dynamics is described by only a single parameter N ~
2/(1-C(l_{min})) where C(l) is a correlation function of the string tangent
vectors. The model predicts an appearance of two new length scales: the
coherence length \xi ~ t/N^2 and the cross-correlation length \chi ~ t/N. At
the onset of evolution N ~ 10 and at late times N is expected to grow
logarithmically due to cosmological stretching and emission of small loops. The
very late time evolution might be modified further when the gravitational
back-reaction scale grows larger than l_{min}.Comment: 5 pages, minor corrections, accepted for publication in Physical
Review
