Invariant Coupling of Determinantal Measures on Sofic Groups

Abstract

To any positive contraction $Q$ on $\ell^2(W)$, there is associated a determinantal probability measure ${\mathbf P}^Q$ on $2^W$, where $W$ is a denumerable set. Let $\Gamma$ be a countable sofic finitely generated group and $G = (\Gamma, \mathsf{E})$ be a Cayley graph of $\Gamma$. We show that if $Q_1$ and $Q_2$ are two $\Gamma$-equivariant positive contractions on $\ell^2(\Gamma)$ or on $\ell^2(\mathsf{E})$ with $Q_1 \le Q_2$, then there exists a $\Gamma$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination ${\bf P}^{Q_1} \preccurlyeq {\bf P}^{Q_2}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when $\Gamma$ 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 ${\bf P}^Q$ as above are ${\bar d}$-limits of finitely dependent processes. Thus, when $\Gamma$ is amenable, ${\bf P}^Q$ is isomorphic to a Bernoulli shift, which was known before only when $\Gamma$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.