Uniqueness of the infinite tree in low-dimensional random forests

Abstract

The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is sampled with probability proportional to β #edges for some β ≥ 0, which arises as the q → 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 Z d , and prove that when d ≤ 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 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 Z d is connected almost surely when d ≤ 4. This proof also provides strong heuristic evidence for the conjecture that the supercritical arboreal gas contains infinitely many infinite trees in dimensions d ≥ 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.

Anomalous criticality coexists with giant cluster in the uniform forest model

“…( 2) can be formulated in the RG framework: the order parameter of the non-linear sigma model in the low-temperature phase has the longitudinal and transverse modes; the nonzero background g 0 comes from the longitudinal mode, and the power-law decaying ∼ |r | 2−d is attributed to the Gaussian fluctuations along the transverse direction. In addition, it was proven [26] that the supercritical phase has, in the infinite-lattice limit, a unique infinite tree for d = 3, 4.…”

“…It was found that w c = 0.43365(2), ν = 1.28(4), and β/ν = 0.4160(6) [25]. Moreover, the supercritical phase also has an infinite and unique tree [26], similar to percolation. This is confirmed in the inset of Fig.…”

Percolation transition for random forests in $d\geqslant 3$

“…As already mentioned, for the arboreal gas we only expect this to be true in d 4. Significant progress towards this statement has been obtained in [58], where it is shown that translation-invariant infinite volume limits of the arboreal gas have a unique infinite tree in d = 3, 4. More precisely, [58] makes use of the existence results of the present paper and establishes uniqueness.…”

“…Significant progress towards this statement has been obtained in [58], where it is shown that translation-invariant infinite volume limits of the arboreal gas have a unique infinite tree in d = 3, 4. More precisely, [58] makes use of the existence results of the present paper and establishes uniqueness. Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas.…”