Ergodic Theory and Dynamical Systems

To any positive contraction Q on `2(W ), there is associated a determinantal probability measure PQ on 2W , where W is a denumerable set. Let 0 be a countable sofic finitely generated group and G = (0, E) be a Cayley graph of 0. We show that if Q1 and Q2 are two 0-equivariant positive contractions on `2(0) or on `2(E) with Q1 ≤ Q2, then there exists a 0-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination PQ1 4 PQ2 . In particular, this applies to the wired and free uniform spanning forests, which was known before only when 0 is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures PQ as above are d̄-limits of finitely dependent processes. Thus, when 0 is amenable, PQ is isomorphic to a Bernoulli shift, which was known before only when 0 is abelian. We also prove analogous results for sofic unimodular random rooted graphs.