Result 231, Probability and statistical mechanics

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 formalization Proof

The bigger picture

Why it matters

The manuscript reports that a globally defined random forest can be generated from independent random labels attached to vertices. The same symmetry-respecting recipe works across a broad class of infinite graphs, without choosing a starting vertex.

What changes?

The free uniform spanning forest is the limiting random, cycle-free edge set obtained from uniformly chosen spanning trees on finite portions of a graph, with boundary vertices left separate. The claimed representation applies to every infinite connected simple unweighted graph with finitely many neighbors per vertex. A single Borel-measurable rule converts independent, identically distributed vertex labels into this forest and respects graph isomorphisms, using no root. This makes the forest a factor of IID.

What does that help mathematicians do?

This would show that the forest's long-range dependencies need no separate source of shared randomness: independent local seeds suffice, though the rule need not be local. The manuscript also reports this representation for every translation-invariant strongly Rayleigh zero-or-one-valued process, a class with strong negative-dependence properties, on every countable group. This includes invariant determinantal processes with Hermitian positive-contraction kernels and requires no amenability assumption on the group.

Are there practical applications?

Its immediate value is foundational: it places these random forests and negatively dependent processes within a common framework of symmetry-preserving constructions from independent noise. For researchers studying random graphs and statistical-mechanical models, that clarifies what randomness is needed. The stated result supplies no runtime or finite-neighborhood guarantee, so it should not be read as an efficient simulation method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The free uniform spanning forest is a factor of IID

September 25, 2026 19 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026,
  author = {{OpenAI}},
  title = {{The free uniform spanning forest is a factor of IID}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf}{OAI:The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/231.md.

The free uniform spanning forest is a factor of IID

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves that the free uniform spanning forest is a factor of independent identically distributed vertex labels on every infinite connected locally finite simple unweighted graph. One Borel equivariant rule works for all such graphs and uses no distinguished root.

The linked supplements also give equivariant IID sampling for invariant strongly Rayleigh laws on countable groups and existence, uniqueness, and IID-factor results for determinantal laws with Hermitian positive-contraction kernels on countable index sets. These group results require neither amenability nor finite generation, and the kernels need not be trace class.

Comparator links

Result Comparator statement
Strongly Rayleigh and determinantal-process IID factors StronglyRayleighDPP.lean
A universal IID factor for the free uniform spanning forest FreeUniformSpanningForest.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.