Factor of iid's through stochastic domination

We develop a method to prove that certain percolation processes on amenable random rooted graphs are factors of iid (fiid), given that the process is a monotone limit of random finite subgraphs that satisfy a certain independent stochastic domination property. Among the consequences are the previously open claims that the Uniform Spanning Forest (USF) is a factor of iid for recurrent graphs, it is a finite-valued finitary fiid on amenable graphs, and that the critical Ising model on $\Z^d$ is a finite-valued finitary fiid, using the known uniqueness of the Gibbs measure.

💡 Research Summary

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The paper introduces a unified probabilistic framework for constructing factor of iid (FiID) representations of a broad class of percolation processes on infinite random rooted graphs. The central notion is that of a compatible monotone weak limit: a random subgraph μ of a unimodular random graph G is said to be a monotone decreasing (or increasing) limit if, for every finite connected graph H, one can define a random subgraph μ_H of H such that (1) μ_H converges weakly to μ when H exhausts G, (2) for any collection of vertex‑disjoint subgraphs H₁,…,H_k of H the product measure ⊗ μ_{H_i} stochastically dominates (or is dominated by) μ_H restricted to the union of the H_i, and (3) μ_H depends only on the isomorphism class of H. This definition captures the classical monotonicity of free versus wired boundary conditions in spanning forests, Ising models, and FK‑random‑cluster models.

Assuming invariant amenability of G (i.e., the existence of a hyperfinite exhaustion {Γ_n} with finite components that is itself a FiID), the authors prove Theorem 4: any random subgraph that is a compatible monotone weak limit is a factor of iid. The construction proceeds inductively over the exhaustion. For each finite component H of Γ_n a coding map φ_n, measurable and equivariant, translates iid labels on H into a sample from μ_H. By Strassen’s theorem, one can couple μ_H with the independent samples on the subcomponents {H_i} in a way that respects the required stochastic domination. This coupling is implemented by locally adding or deleting edges based only on the labels inside H, guaranteeing that each edge changes status at most twice. Consequently the limit ω_G = lim_n ω_{Γ_n} exists almost surely and is a FiID.

Applying this abstract result yields several concrete corollaries:

1. Uniform Spanning Forest (USF). For recurrent unimodular random graphs the free and wired USFs coincide, and both are compatible monotone limits (free decreasing, wired increasing). By Theorem 4 the USF is a FiID; moreover, using the hyperfinite exhaustion with finite‑valued labels, Theorem 9 shows that the USF is a finite‑valued finitary FiID on any invariantly amenable graph.

2. Ising Model at Criticality. The set of plus‑spins under all‑plus boundary conditions forms a compatible monotone decreasing limit, while minus‑spins give an increasing limit. When the Gibbs measure is unique (as is known for the critical Ising model on ℤⁿ), the same construction provides a finite‑valued finitary FiID representation (Theorem 7). The authors note that in the presence of multiple Gibbs measures a finitary coding is impossible, consistent with earlier work.

The paper emphasizes the Γ_n‑locality of the constructions: each step uses only the iid labels inside the current finite component, making the algorithms naturally parallelizable and suitable for distributed computation. The transition from arbitrary