Indistinguishability of collections of trees in the uniform spanning forest
We prove the following indistinguishability theorem for k-tuples of trees in the uniform spanning forest of Z^d: Suppose that A is a property of a k-tuple of components that is stable under finite modifications of the forest. Then either every k-tuple of distinct trees has property A almost surely, or no k-tuple of distinct trees has property A almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.
Additional Information© 2020 Institut Henri Poincaré. Received: 16 October 2018; Revised: 19 February 2019; Accepted: 25 March 2019; Published: May 2020. First available in Project Euclid: 16 March 2020. This work took place while the author was an intern at Microsoft Research, Redmond.
Submitted - 1810.06382.pdf