# Dynamics & probability: Asaf Nachmias (Tel Aviv)

Title: Indistinguishability of trees in uniform spanning forests Abstract: The uniform spanning forest (USF) of an infinite connected graph G is the weak limit of the uniform spanning tree measure taken on exhausting finite subgraphs of G. It is easy to see that it is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when G=Zd, the USF is almost surely a connected tree if and only if d=1,2,3,4. We prove that when G is a Cayley graph (or more generally, a unimodular random network) one cannot distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm 2001. Joint work with Tom Hutchcroft.

## Date:

Tue, 03/11/2015 - 14:00 to 15:00

## Location:

Manchester building, Hebrew University of Jerusalem, (Room 209)