On Negative Correlation of Arboreal Gas on Some Graphs ^††

Let $\mathbb{P}^{\rm perc}_{p}$ denote the probablity of Bernoulli bond percolation with parameter $p$. We can note that Arboreal Gas with a uniform parameter $\beta$ is equivalent to Bernoulli bond percolation with parameter $p_{\beta}:=\beta/(1+\beta)$ conditioned to be acyclic:

Another notable observation is that Arboreal Gas emerges as the limit of the $q$-states random cluster model as $q\to 0$, with $p=\beta q$, as mentioned in [13]. The uniform forest model, also mentioned in [13], can be seen as one particular instance of Arboreal Gas, where $\beta=1$. It is important to note that this model is distinct from another known model referred to as uniform spanning forest in the literature. Furthermore, another specific case is the uniform spanning tree, which arises as a limit of Arboreal Gas as $\beta\to\infty$.

An interesting phenomenon lies in the correlation between Arboreal Gas and hyperbolic spin systems. This kind of spin system is different from the classical spin systems with spherical symmetry, such as Ising model and classical Heisenberg model. Hyperbolic spin systems are also extensively studied in condensed matter physics due to their connection to the Anderson delocalisation-localisation transition of random Schödinger operators and related matrix models, as discussed in [7, 14, 15]. Researchers are interested in the existence of phase transitions in the hyperbolic spin systems. Considering Arboreal Gas, Bauerschmidt, Crawford, Helmuth and Swan presented a magic formula in [3] that elegantly describe its probability by the intergral in the hyperbolic spin system $\mathbb{H}^{0|2}$, which may also be seen in [1, 6].

The subcritical percolation and percolating phase transitions in Arboreal Gas have garnered significant attention. In the seminal works [4, 10, 12], it was established that Arboreal Gas, with a parameter $\beta=\alpha/N$ for fixed $\alpha$, undergoes a phase transition on the complete graph $K_{N}$ with $N$ vertices. This phase transition was also demonstrated by Bauerschmidt, Crawford, Helmuth, and Swan in their recent study [3]. Additionally, they proved the polynomial decay of connectivity probability from the origin to other vertices in all subgraphs of the lattice $\mathbb{Z}^{2}$. Based on this polynomial decay, and assuming for the existence of a weak limit of probabilities on asymptotic graphs of $\mathbb{Z}^{2}$, they demonstrated the absence of an infinite tree almost surely. On torus $\Lambda_{N}=\mathbb{Z}^{d}/L^{N}\mathbb{Z}^{d}$ with $L$ fixed and $N\to\infty$, Bauerschmidt, Crawford and Helmuth gave an asymptotic estimate of connectivity probability from the origin to other vertices for $d\geq 3$ in [2]. Recently, Halberstam and Hutchcroft verified the uniqueness of infinite tree in $\mathbb{Z}^{d}$ for $d=3,4$ for any translation-invariant Arboreal Gas Gibbs measures in [9].

For any edge set $S\subset E$, let $\mathbb{P}_{\beta}[S]$ be the probability that all edges in $S$ belong to the forest. We simply write $\mathbb{P}_{\beta}[S]$ as $\mathbb{P}_{\beta}[e_{1}e_{2}\dots e_{n}]$ for $S=\{e_{1},e_{2},\dots,e_{n}\}$. For two vertices $u,v\in V$, denote by $\mathbb{P}_{\beta}[u\leftrightarrow v]$ the probability that $u,v$ are in a same tree of the forest. The negative correlation of Arboreal Gas should be expressed as

Until now, this question is still open. A weaker inequality has been proved recently in [5] by Brändén and Huh. They showed $\mathbb{P}_{\beta}[e_{1}e_{2}]\leq 2\mathbb{P}_{\beta}[e_{1}]\mathbb{P}_{\beta}[e_{2}]$ by using the Lorentzian signature. Once (1.1) holds, the existence of the weak limit of Arboreal Gas Gibbs measures on increasing asymptotic graphs $G_{n}\uparrow\mathbb{Z}^{2}$ as $n\to\infty$ was answered in [3]. Under this weak limit on $\mathbb{Z}^{2}$, all trees should be finite almost surely, see Corollary 1.4 in [3].

In this paper, we focus on the negative correlation of Arboreal Gas on finite connected graph $G$. One way to consider this question is to realize the relevance between Arboreal Gas and the hyperbolic spin system $\mathbb{H}^{0|2}$. The negative correlation of Arboreal Gas can be implied by the monotonicity of a hyper-integral on $\mathbb{H}^{0|2}$, see section 5.2 in [1]. However, the monotonicity of hyper-integral is hard to be verified. In our paper, we consider Arboreal Gas directly. Here is the outline of this paper. In Section 2, we show our main results, and we give some preliminaries in Section 3. In Section 4, we show the negative correlation for adjacent edges and sufficiently large $\beta$. In Section 5, we show the negative correlation of complete graphs $K_{n}$ with $n$ vertices for sufficiently large $n$ and $\beta$. In Section 6, we give the proof of the equivalence of the negative correlation on any graph and on its simplified version.

We verify those two theorems by regarding the probability $\mathbb{P}_{\beta}[e_{1}e_{2}],\mathbb{P}_{\beta}[e_{1}],\mathbb{P}_{\beta}[e_{2}]$ as polynomials of $\beta$. A fact is, the negative correlation holds for uniform spanning trees on finite connected graphs, see section 4.2 in [11]. An important observation is that the term with highest degree in the polynomials of $\beta$ mentioned before corresponds to the uniform spanning trees. This is our basic idea to show the negative correlation for sufficiently large $\beta$.

Besides, we have another theorem to simplify the structure of graphs when we consider the negative correlation. Before we give this theorem, we first give some definitions.

Here we give our main theorem.

This corollary immediately comes out by Theorem 2.6. We can also obtain the negative correlation on some specific graphs.

Secondly, we prove that the decreasing of $\mathbb{P}_{\beta}[e_{0}]$ in each $\beta_{e}$ for all $e_{0}\in E$ and $e\neq e_{0}$ deduces condition $2$, which may complete the proof of $1\Rightarrow 2$. To show this property, we claim that

$\mathbb{P}_{\beta}[\cdot|S]$ equals the limit of $\mathbb{P}_{\beta}[\cdot]$ as $\beta_{e}\to\infty$ for all $e\in S\subset E$.

By the decreasing of $\mathbb{P}_{\beta}[e_{0}]$ in each component of $\beta$ except $\beta_{e_{0}}$ for all $e_{0}\in E$, we know that for $S\subset E$,

For disjoint $S_{1},S_{2}\subset E$, Applying this to $\mathbb{P}[\cdot|S_{1}]$, we obtain

This inequality is equivalent to

BY induction of $S_{1}$, we verify that

which implies condition $2$.

Finally we prove our claim by setting

Then $\mathbb{P}_{\beta^{x,S}}[F]=\prod_{e\in F}\beta_{e}^{x,S}/\sum_{F^{\prime}\in\mathcal{F}}\prod_{e\in F^{\prime}}\beta_{e}^{x,S}$, which converges to (as $x\to\infty$)

By $\mathbb{P}_{\beta}[F|S]\propto\prod_{e\in F\setminus S}\beta_{e}\textbf{1}_{\{F\supset S\}}$, we show that $\mathbb{P}_{\beta^{x,S}}[\cdot]$ converges to $\mathbb{P}_{\beta}[\cdot|S]$ in weak topology. Further, by the decreasing of $\mathbb{P}_{\beta^{x,S}}[\cdot]$ in $x$, we conclude that $\mathbb{P}_{\beta}[\cdot|S]\leq\mathbb{P}_{\beta}[\cdot]$.  

Next we show the negative correlation of uniform spanning trees. Here we give the definition of the uniform spanning trees.

In our next part, denote by $\mathbb{P}_{\infty}^{G}$ the probability of uniform spanning trees on graph $G$. In fact, uniform spanning trees can be generated by simple random walks. This method is called Wilson’s algorithm, see the details of which in Theorem 4.1 of [11]. To show the negative correlation of uniform spanning trees, we use this method and another tool called electrical network. This tool shows a nice relationship between the probability theory and potential theory. Here we define the electrical network.

For a finite connected graph $G$ with conductance $c_{e}$ on each edge, we build an electrical network. Given two disjoint vertex sets $S$ and $T$ to be source and sink. Define potential difference $\phi(\vec{e})$ and current $i(\vec{e})$ on each direct edge $\vec{e}$. For adjacent vertices $u,v$,

Here we normally take

Further, the potential difference and current satisfies the following three laws, see [8].

For some finite connected graph $G$, we have the following equation for its spanning trees.

By Kirchhoff’s Effective Resistance Formula, we deduce the negative correlation of uniform spanning trees. Before that, we give a lemma to show that the effective resistance decreases if the resistance on some edge decreases by the following lemma.

If $i(\vec{e_{0}})=0$, we have $E_{c^{\prime}}(i)=E_{c}(i)$. Here we prove that $i$ is the unit current flow on the electrical network with conductance $c^{\prime}$. Consider the potential $\phi$ on the electrical network with conductance $c$, then for any edge $e\in E$,

This implies that $\phi(\vec{e_{0}})=0$. On the electrical network with conductance $c^{\prime}$, $\phi$ and $i$ satisfy Kirchhoff’s potential law and Kirchhoff’s current law. As to Ohm’s law, for $e\neq e_{0}$,

For $e=e_{0}$,

Therefore, $i$ is a unit current flow on the electrical network with conductance $c^{\prime}$, which implies

In Case (a), if we start a unit current flow $i$ from $e_{1}^{-}$ to $e_{1}^{+}$, then $i(\vec{e_{2}})$ is not vanishing.

In Case (b), we prove that $e_{1},e_{2}$ can be separated into two parts of the graph $G$, where the intersection of these two parts has only one vertex. By this property, the probability of Arboreal Gas on $G$ can be expressed as the product of the probabilities of Arboreal Gas on these two parts.

Now we prove Theorem 2.1.

Now we prove the claim by contradiction. Otherwise, there is a simple cycle $C=u_{1}u_{2}\dots u_{n}u_{n+1}$ with $u_{n+1}:=u_{1}$ such that $u_{k_{i}}u_{k_{i}+1}\in E_{i}$ for $i=1,2$ and some $k_{1}\neq k_{2}$. Without loss of generality, assume $1\leq k_{1}<k_{2}\leq n$. Since $E_{1}\cup E_{2}=E$, there should be two distinct integers $n_{1},n_{2}$ with $k_{1}\leq n_{1}<k_{2}$ and $n_{2}<k_{1}$ or $n_{2}\geq k_{2}$ such that $u_{n_{1}}u_{n_{1}+1},u_{n_{2}+1}u_{n_{2}+2}\in E_{1}$ and $u_{n_{1}+1}u_{n_{2}+2},u_{n_{2}}u_{n_{2}+1}\in E_{2}$. This implies $u_{n_{1}+1},u_{n_{2}+1}\in V_{1}\cap V_{2}$, i.e., $u_{n_{1}+1}=u_{n_{2}+1}=e_{1}^{+}$, which is contradict to $n_{1}\neq n_{2}$ and that $C$ is a simple cycle.  

Finally we prove Lemmas 4.1 and 4.2.

For integer $k\geq 1$, assume there is a chain $v_{0}v_{1}\dots v_{k}$ with $i(\overrightarrow{v_{j-1}v_{j}})<0$ for $j=1,\dots,k$ and $v_{j_{1}}\neq v_{j_{2}}\neq e_{1}^{+}$ for distinct $j_{1},j_{2}=1,\dots,k$. By Kirchhoff’s current law, we can find a vertex $v_{k+1}$ adjacent to $v_{k}$ with $i(\overrightarrow{v_{j-1}v_{j}})<0$. Furthermore, $v_{k+1}\neq v_{j},e_{1}^{+}$ for $j=0,\dots,k$. Otherwise, if $v_{k+1}=e_{1}^{+}$, potentials on the cycle $v_{0}v_{1}\dots v_{k}e_{1}^{-}u_{1}\dots u_{m-1}v_{0}$ do not obey Kirchhoff’s potential law. if $v_{k+1}=v_{j}$ for some $j=0,1,\dots,k$, potentials on the cycle $v_{j}v_{j+1}\dots v_{k}v_{j}$ do not obey Kirchhoff’s potential law.