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Markovian linearization of random walks on groups

In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra A to the study of a polynomial of degree one, evaluated on the enlarged algebra A x M r (C), for some integer r. We introduce a new instance of the linearization trick which is tailored to study a finitely supported random walk on a group G by studying instead a nearest-neighbor colored random walk on G x {1,. .. , r}, which is much simpler to analyze. As an application we extend well-known results for nearest-neighbor walks on free groups and free products of finite groups to colored random walks, thus showing how one can obtain explicit formulas for the drift and entropy of a finitely supported random walk.