The free uniform spanning forest is a factor of IID

On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID. (Lean)

Manuscript

The free uniform spanning forest is a factor of IID

September 25, 2026 · source · cite as \cite{OAI:The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026}

We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.

Result 231 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.