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We investigate the limit behaviour of random walks on some non-commutative discrete groups related to knot theory. Namely, we study the connection between the limit behaviour of the Lyapunov exponent of products of non-commutative random matrices-generators of the braid group-and the asymptotics of powers of the algebraic invariants of randomly generated knots. We turn the simplest problems of knot statistics into the context of random walks on hyperbolic groups. We also consider the limit distribution of Brownian bridges on so-called locally non-commutative groups.