Abstract
We investigate numerically and analytically the statistics of Markov chains on so-called braid (B n ) and locally free (ℒℱ n ) groups. Namely, we compute the mean length 〈μ〉 and the variance 〈μ2〉−〈μ〉2 of the shortest word which remains after applying of all group relations to the randomly generatedN-letter word (Markov chain). We express the conjecture (numerically justified) that the mean value 〈μ〉 for the random walk on the groupB n (n≫1) coincides with high accuracy with the same value for the random walk on the “locally free group weth errors” if the number of errors is of order of 20%.
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Desbois, J., Nechaev, S. Statistical mechanics of braided Markov chains: I. Analytic methods and numerical simulations. J Stat Phys 88, 201–229 (1997). https://doi.org/10.1007/BF02508470
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DOI: https://doi.org/10.1007/BF02508470