[Paper Review] Uniqueness of the infinite tree in low-dimensional random forests

Noah Halberstam, Tom Hutchcroft|arXiv (Cornell University)|Feb 23, 2023

Stochastic processes and statistical mechanics39 references4 citations

TL;DR

This paper establishes that in low-dimensional hypercubic lattices (d ≤ 4), any translation-invariant Gibbs measure for the arboreal gas model contains at most one infinite tree almost surely. The proof combines a novel resampling property for arboreal gas measures with a new result showing that the uniform spanning forest on any translation-invariant random connected subgraph of ℤ^d is connected a.s. when d ≤ 4, thereby resolving the uniqueness of the infinite tree in this regime.

ABSTRACT

The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is sampled with probability proportional to $β^{\# ext{edges}}$ for some $β\geq 0$, which arises as the $q o 0$ limit of the Fortuin-Kastelyn random cluster model with $p=βq$. We study the infinite-volume limits of the arboreal gas on the hypercubic lattice $\mathbb{Z}^d$, and prove that when $d\leq 4$, any translation-invariant infinite volume Gibbs measure contains at most one infinite tree almost surely. Together with the existence theorem of Bauerschmidt, Crawford and Helmuth (2021), this establishes that for $d=3,4$ there exists a value of $β$ above which subsequential weak limits of the $β$-arboreal gas on tori have exactly one infinite tree almost surely. We also show that the infinite trees of any translation-invariant Gibbs measure on $\mathbb{Z}^d$ are one-ended almost surely in every dimension. The proof has two main ingredients: First, we prove a resampling property for translation-invariant arboreal gas Gibbs measures in every dimension, stating that the restriction of the arboreal gas to the trace of the union of its infinite trees is distributed as the uniform spanning forest on this same trace. Second, we prove that the uniform spanning forest of any translation-invariant random connected subgraph of $\mathbb{Z}^d$ is connected almost surely when $d\leq 4$. This proof also provides strong heuristic evidence for the conjecture that the supercritical arboreal gas contains infinitely many infinite trees in dimensions $d\geq 5$. Along the way, we give the first systematic and axiomatic treatment of Gibbs measures for models of this form including the random cluster model and the uniform spanning tree.

Motivation & Objective

• To resolve the long-standing open question of whether the infinite-volume limit of the arboreal gas on ℤ^d contains a unique infinite tree in low dimensions.
• To establish that for d ≤ 4, any translation-invariant Gibbs measure of the arboreal gas contains at most one infinite tree almost surely.
• To provide a rigorous axiomatic framework for Gibbs measures in models like the random cluster model and uniform spanning tree.
• To offer strong heuristic evidence that the supercritical arboreal gas contains infinitely many infinite trees in d ≥ 5.
• To prove that infinite trees in any translation-invariant Gibbs measure on ℤ^d are one-ended almost surely in all dimensions.

Proposed method

• Introduces a resampling property for translation-invariant arboreal gas Gibbs measures, showing that the restriction of the measure to the trace of its infinite trees is distributed as the uniform spanning forest on that trace.
• Proves that the uniform spanning forest of any translation-invariant random connected subgraph of ℤ^d is connected almost surely when d ≤ 4.
• Applies a criterion for the infinite intersection property of random walks in unimodular random graphs, relying on Green's function estimates and volume growth bounds.
• Uses dyadic decomposition of space and Green's function comparisons across scales to bound the expected number of intersections between two independent random walks.
• Applies Fatou’s lemma and Markov-type inequalities to show that the total expected number of intersections is infinite under the given conditions.
• Leverages the Varopoulos-Carne inequality and diffusive displacement estimates to extend results to unimodular random graphs with polynomial volume growth.

Experimental results

Research questions

• RQ1 Does the infinite-volume limit of the arboreal gas on ℤ^d contain a unique infinite tree when d ≤ 4?
• RQ2 Is the uniform spanning forest on a translation-invariant random connected subgraph of ℤ^d almost surely connected when d ≤ 4?
• RQ3 Can the resampling property for arboreal gas Gibbs measures be established without symmetry assumptions?
• RQ4 What is the structure of the infinite tree(s) in the arboreal gas, particularly regarding their end structure?
• RQ5 Does the supercritical arboreal gas contain infinitely many infinite trees in dimensions d ≥ 5?

Key findings

• For d ≤ 4, any translation-invariant Gibbs measure of the arboreal gas on ℤ^d contains at most one infinite tree almost surely.
• The uniform spanning forest of any translation-invariant random connected subgraph of ℤ^d is connected almost surely when d ≤ 4.
• The infinite trees of any translation-invariant Gibbs measure on ℤ^d are one-ended almost surely in every dimension.
• The proof provides strong heuristic evidence that the supercritical arboreal gas contains infinitely many infinite trees in dimensions d ≥ 5.
• For d ≤ 4, there exists β₀(d) > 0 such that subsequential weak limits of the β-arboreal gas on d-dimensional tori contain exactly one infinite tree almost surely when β > β₀(d).
• The resampling property holds for all translation-invariant arboreal gas Gibbs measures in every dimension, stating that the restriction to the trace of infinite trees is distributed as the uniform spanning forest on that trace.