Zero-temperature ising spin dynamics on the homogeneous tree of degree three

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Is ing model on the homogeneous tree of degree three (T) with random (i.i.d. B ernoulli) spin configuration at time 0. Letting theta denote the probabilit y that any particular vertex has a + 1 initial spin, for theta = 1/2, infin ite spin clusters do not exist at time 0 but we show that infinite 'spin ch ains' (doubly infinite paths of vertices with a common spin) exist in abund ance at any time epsilon > 0. We study the structure of the subgraph of T g enerated by the vertices in time-epsilon spin chains. We show the existence of a phase transition in the I + 1 spin chains almost surely never form se nse that for some critical theta (c) with 0 < <theta>(c) 1/2, +1 for theta < <theta>(c) but almost surely do form in finite time for theta > theta (c) . We relate these results to certain quantities of physical interest, such as the t --> infinity asymptotics of the probability p(t) that any particul ar vertex changes spin after time t.