Arboreal Gas Measure in Spanning Forests

• Arboreal gas measure is a probability on spanning forests where each forest is weighted by β per edge and constrained to be acyclic.
• It is equivalent to Bernoulli bond percolation conditioned on acyclicity and represents the q→0 limit of the Fortuin–Kasteleyn random cluster model.
• In infinite volumes, the model reveals phase transitions, connectivity properties, and resampling identities linked to uniform spanning forests.

The arboreal gas measure is a probability measure on unrooted spanning forests of a graph in which each forest is weighted by a factor $\beta$ per edge. On finite graphs it is equivalent to Bernoulli bond percolation with parameter $p=\beta/(1+\beta)$ conditioned on acyclicity, and it arises as the $q\to 0$ limit of the Fortuin–Kasteleyn random cluster model with $p=\beta q$. In infinite volume it is studied through weak limits and Gibbs measures, with current theory focusing on phase transitions, the geometry of infinite trees, resampling identities linked to uniform spanning forests, and unresolved correlation inequalities (Halberstam et al., 2023, Bauerschmidt et al., 2021).

1. Finite-volume definition and basic variants

Let $G=(V,E)$ be a finite undirected graph. A spanning forest of $G$ is an acyclic spanning subgraph $F\subseteq G$, and the finite-volume arboreal gas with parameter $\beta\ge 0$ assigns probability

$$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$

to each spanning forest $F$, where $p=\beta/(1+\beta)$0 is the number of occupied edges and $p=\beta/(1+\beta)$1 is the set of all spanning forests of $p=\beta/(1+\beta)$2. A more general form allows edge-dependent weights $p=\beta/(1+\beta)$3 with

$p=\beta/(1+\beta)$4

The model is therefore a forest measure with no distinguished roots and with all combinatorial structure entering through the acyclicity constraint and the edge fugacities (Huang, 2023).

For finite subgraphs $p=\beta/(1+\beta)$5, boundary conditions are encoded by an equivalence relation $p=\beta/(1+\beta)$6 on the boundary $p=\beta/(1+\beta)$7. A forest $p=\beta/(1+\beta)$8 is said to extend $p=\beta/(1+\beta)$9 when the quotient graph $q\to 0$0 remains acyclic, and the corresponding finite-volume specification is

$q\to 0$1

This boundary formalism is the finite-volume mechanism that later supports the infinite-volume Gibbs theory on $q\to 0$2 (Halberstam et al., 2023).

The limit $q\to 0$3 is a distinguished case. Conditioning on acyclicity then forces maximal spanning forests, and when the relevant quotient graph is connected the model becomes the usual uniform spanning tree. In this sense, the arboreal gas interpolates between sparse forest ensembles and uniform spanning-tree or spanning-forest measures (Halberstam et al., 2023).

2. Infinite-volume Gibbs measures and augmented connectivity

On infinite graphs, and in particular on $q\to 0$4, the arboreal gas is defined through subsequential weak limits of finite-volume measures along exhaustions. Halberstam and Hutchcroft formulate this in an augmented DLR-like framework because ordinary quasilocal specifications fail for connectivity-based models: conditional laws inside a finite region depend on how vertices are connected through the complement, including “through infinity” (Halberstam et al., 2023).

The basic object is an augmented subgraph $q\to 0$5, where $q\to 0$6 is a subgraph and $q\to 0$7 is a boundary map assigning to each finite subgraph $q\to 0$8 an equivalence relation $q\to 0$9 on $p=\beta q$0. This records which boundary vertices are connected outside $p=\beta q$1. The corresponding augmented connectivity relation is

$p=\beta q$2

A probability measure $p=\beta q$3 on spanning forests is called a $p=\beta q$4-arboreal gas Gibbs measure if it arises as the projection of an augmented measure $p=\beta q$5 for which, for every finite $p=\beta q$6,

$p=\beta q$7

This gives an axiomatic infinite-volume specification that accommodates long-range connectivity information (Halberstam et al., 2023).

A structural result identifies these Gibbs measures exactly with subsequential weak limits of finite-volume arboreal gas measures on exhaustions with arbitrary, possibly random, boundary conditions. In the translation-invariant setting on $p=\beta q$8, the augmented formalism also yields ergodicity and tail-triviality statements for extremal Gibbs measures. A further corollary states that for translation-invariant augmented Gibbs measures on $p=\beta q$9, the augmentation is almost surely the wired augmentation of the forest, so the boundary data are not an independent extra degree of freedom (Halberstam et al., 2023).

3. Relation to percolation, random-cluster theory, spanning forests, and the $G=(V,E)$0 model

The finite-volume arboreal gas is precisely Bernoulli bond percolation with edge-open probability

$G=(V,E)$1

conditioned on the resulting subgraph being acyclic. This representation is exact on finite graphs and gives the simplest probabilistic interpretation of the model: one starts from i.i.d. bond percolation and discards all configurations containing cycles (Huang, 2023).

It is also the $G=(V,E)$2 scaling limit of the Fortuin–Kasteleyn random cluster model with $G=(V,E)$3. In that limit, cyclic configurations acquire negligible weight compared with forests, and the remaining weights are proportional to $G=(V,E)$4. This places the arboreal gas in the same formal family as FK and Potts models, but at the singular $G=(V,E)$5-state end of the random-cluster parameter range (Halberstam et al., 2023).

At $G=(V,E)$6, the model coincides with the uniform maximal spanning forest, and on $G=(V,E)$7 the infinite-volume UST/WUSF is a $G=(V,E)$8 arboreal gas Gibbs measure. The resampling results discussed below make this relationship more than formal: the arboreal gas restricted to its infinite locus behaves as a wired uniform spanning forest on that random trace (Halberstam et al., 2023).

Bauerschmidt, Crawford, Helmuth, and collaborators connect the model to a non-linear sigma model with target space the fermionic hyperbolic plane $G=(V,E)$9. In that formulation, the arboreal gas is the random-cluster representation of a $G$0-state Potts-type model with continuous symmetries. The starting point of the high-dimensional analysis is an exact relationship between the arboreal gas and this $G$1 model; Ward identities and renormalisation-group methods then link symmetry breaking in the field theory to the existence of infinite trees in the forest model (Bauerschmidt et al., 2021).

4. Phase structure on regular trees and on $G$2

On the infinite $G$3-regular tree with wired boundary conditions, the weak limit of the arboreal gas exists along any exhaustion and does not depend on the exhaustion. The model undergoes a phase transition at

$G$4

Below and at criticality, the wired arboreal gas coincides exactly with bond percolation with parameter $G$5. Above criticality, it is characterised as the superposition of critical bond percolation and a random collection of infinite one-ended paths. This yields a transparent example of a supercritical phase that still exhibits critical-like behaviour, because finite clusters retain critical-percolation laws while infinite components appear only as thin rays rather than as ordinary supercritical branching clusters (Easo, 2021).

On $G$6, Bauerschmidt, Crawford, and Helmuth proved that the arboreal gas undergoes a percolation phase transition in dimensions $G$7, in contrast with $G$8, where no percolation transition occurs. More precisely, for $G$9 there exists $F\subseteq G$0 such that for every $F\subseteq G$1, subsequential weak limits of the $F\subseteq G$2-arboreal gas on large $F\subseteq G$3-dimensional tori contain at least one infinite tree almost surely, whereas in $F\subseteq G$4 any subsequential limit has no infinite trees for any finite $F\subseteq G$5 (Bauerschmidt et al., 2021).

The mechanism in dimensions $F\subseteq G$6 is expressed in the $F\subseteq G$7 representation as spontaneous breaking of continuous symmetry at low temperatures. In arboreal-gas language, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit, together with massless free-field correlations at low temperatures and the existence of a macroscopic tree on finite tori (Bauerschmidt et al., 2021).

5. Infinite trees, one-endedness, and the resampling principle

The central low-dimensional structural theorem is due to Halberstam and Hutchcroft: when $F\subseteq G$8, any translation-invariant infinite-volume Gibbs measure for the arboreal gas on $F\subseteq G$9 contains at most one infinite tree almost surely. Together with the existence theorem of Bauerschmidt, Crawford, and Helmuth, this implies that for $\beta\ge 0$0 there exists a value of $\beta\ge 0$1 above which subsequential weak limits of the $\beta\ge 0$2-arboreal gas on tori have exactly one infinite tree almost surely (Halberstam et al., 2023).

A second general theorem states that every infinite tree of a translation-invariant arboreal gas Gibbs measure on $\beta\ge 0$3 is one-ended almost surely in every dimension. Thus, even when multiple infinite trees are present, bi-ended and multi-ended infinite trees are excluded in the translation-invariant Gibbs setting (Halberstam et al., 2023).

The key technical bridge to uniform spanning forests is the resampling property. If $\beta\ge 0$4 is sampled from a translation-invariant arboreal gas Gibbs measure and $\beta\ge 0$5 is the set of vertices lying in infinite components, then the induced trace $\beta\ge 0$6 is connected almost surely, and the conditional law of the restriction of $\beta\ge 0$7 to this trace, given $\beta\ge 0$8 and the configuration outside the trace, is the wired uniform spanning forest on $\beta\ge 0$9. Equivalently, one may delete all edges in the infinite trees and resample them according to WUSF on the same trace without changing the law of the full configuration (Halberstam et al., 2023).

The second ingredient is a connectivity theorem for WUSF on translation-invariant random subgraphs: if $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$0, then the wired uniform spanning forest of each infinite connected component of any translation-invariant random connected subgraph of $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$1 is connected almost surely. Combined with the resampling property, this yields uniqueness of the infinite arboreal-gas tree in low dimensions and provides strong heuristic evidence that in dimensions $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$2 the supercritical arboreal gas should contain infinitely many infinite trees (Halberstam et al., 2023).

6. Negative correlation, partial results, and outstanding problems

A central unresolved problem is negative correlation. For edges $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$3, the conjectured inequality is

$$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$4

equivalently $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$5 for the edge indicators $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$6. This is not known in full generality for arboreal gas on arbitrary finite graphs, and the existence and behaviour of weak limits on infinite graphs is tied to this question (Huang, 2023).

Brändén and Huh proved a weaker bound,

$$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$7

using Lorentzian polynomials. The sharp inequality remains open. The same paper establishes that pairwise negative correlation is equivalent to a stronger set-wise inequality for disjoint edge sets, and also to a monotonicity statement asserting that the marginal probability of a fixed edge is decreasing in the weight of every other edge (Huang, 2023).

Several partial results are known. Negative correlation holds for adjacent edges in the large-$$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$8 regime on finite connected graphs, for complete graphs $$\mathbb{P}_\beta(F)=\frac{1}{Z_\beta}\,\beta^{|F|}, \qquad Z_\beta=\sum_{F\in\mathcal F(G)}\beta^{|F|},$$9 with sufficiently large $F$0 and $F$1 sufficiently large or sufficiently small, and for ladder graphs for all edge-parameters. The proofs combine spanning-tree asymptotics, electrical network arguments such as effective resistance and Rayleigh’s principle, Kirchhoff’s matrix-tree theorem, and graph-simplification procedures based on deleting pivotal edges, reducing degree-2 vertices, and merging multiple edges (Huang, 2023).

The importance of this problem is not merely local. Bauerschmidt, Crawford, Helmuth, and Swan show that once negative correlation is available, one can establish tightness and weak-limit existence for arboreal gas measures on increasing asymptotic graphs of $F$2; in those weak limits all trees are finite almost surely. More broadly, a complete negative-correlation theory would sharpen current understanding of infinite-volume existence, monotonicity in $F$3, and phase structure across graph classes and dimensions (Huang, 2023).