ABSTRACT Let G be the vertex set of a homogeneous, infinite tree of finite degree. Let {RXy)x ye c be the transition matrix of a random walk on G which proceeds along edges of the tree. Suppose that R is a symmetric matrix, so that there is a probability assigned to each undirected edge. Suppose also that the walk is vertex homogeneous, i.e. that the probabilities assigned to the edges leaving a given vertex form a multiset which is independent of the vertex. The group G — Z2 * Z2 * • • • • Z2 acts simply transitively on G, leaving the random walk invariant. Thus R commutes with the action of G on £2(G). For a e sp(i2) we construct H a , the generalized cr-eigenspace of R, which is a representation space for G. We analytically continue these principal series representations to find complementary series representations. Our main result is that these representations are (with a few exceptions) irreducible. Key words and phrases, homogeneous tree, free group, algebraicity, nearest-neighbor, ran- dom walk, unitary representation, principal series, complementary series, boundary realization, irreducibility, inequivalence.

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