Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 1 / b

Let ${\mu }_n$ be the uniform spanning tree measure on the finite connected induced graph $G_n$. Uniform spanning-tree edge indicators are negatively associated: increasing events depending on disjoint edge sets have nonpositive covariance. Together with the spatial Markov property, this implies the free-boundary monotonicity under graph enlargement: for every increasing cylinder event $A$, ${\mu }_n(A)$ is eventually nonincreasing once $G_n$ contains all edges on which $A$ depends.

Define the free uniform spanning forest measure byfor increasing cylinder events. These limits determine a unique probability measure, independent of the exhaustion; equivalently, ${\mu }_n$ converges weakly to ${\mu }_F$ on the product space.