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 ${\mathbf{P}}^{Q_1}\preccurlyeq {\mathbf{P}}^{Q_2}$. 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 ${\mathbf{P}}^Q$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when $\Gamma$ is amenable, ${\mathbf{P}}^Q$ 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.

39 pages, no figures; v2 final version

References in corpus (4)

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Cited by in corpus (5)

• Interlacements and the Wired Uniform Spanning Forest
• Random interlacement is a factor of i.i.d
• Radial Toeplitz operators and invariant determinantal point processes on Cayley trees
• Limiting entropy of determinantal processes
• Local weak${}^{\ast }$-Convergence, algebraic actions, and a max-min principle