Statistical mechanics of braided Markov chains .1. Analytic methods and numerical simulations

We investigate numerically and analytically the statistics of Markov chains on so-called braid (B-n) and locally free (LFn) groups. Namely, we compute the mean length [mu] and the variance [mu(2)]-[mu](2) of the shortest word which remains after applying of all group relations to the randomly generated N-letter word (Markov chain). We express the conjecture (numerically justified) that the mean value [mu] for the random walk on the group B-n (n much greater than 1) coincides with high accuracy with the same value for the random walk on the ''locally free group with errors'' if the number of errors is of order of 20%.