Statistical properties of locally free groups with applications to braid groups and growth of random heaps

The main statistical characteristics of locally free groups: the growth, the drift and the entropy are considered and relations between them are established. Our results assert that: (i) the statistical properties of random walks (Markov chains) on locally free and braid groups are not the same as the uniform statistics on these groups, and (ii) the stabilization of the statistical characteristics exists when the number of generators of the group grows.