Markovian Linearization of Random Walks on Groups

Charles Bordenave, Bastien Dubail

Abstract:

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