# Infinite clusters for loopless percolation

I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is appreciated.

But let's consider a percolation model on $$\mathbb{Z}^d$$ (or even $$\mathbb{Z}^2$$ if that's easier), with each open edge contributing weight $$p \in (0, 1)$$, with the condition that no loops can be formed by the open edges. More formally, let $$E$$ be the edges of the grid graph, and $$\mathbf{\omega} \in \{ 0, 1 \}^{|E|}$$ be an edge configuration, then define the measure of the edge configurations to be $$\phi_p(\mathbf{\omega}) = \mathcal{Z}^{-1} p^{o(\mathbf{\omega})}(1-p)^{c(\mathbf{\omega})}\mathbf{1}_A(\mathbf{\omega})$$

where $$o(\mathbf{\omega})$$ is the number of open edges of $$\mathbf{\omega}$$, $$c(\mathbf{\omega})$$ is the number of closed edges, and $$A$$ is the event that there is no cycle within the graph generated by the open bonds of $$\mathbf{\omega}$$. ($$\mathcal{Z}$$ is just some normalizing constant, which would not be 1 in this case.)

The motivation behind this model is that it serves as a highly simplified version of a certain random cluster representation of Ising spin glasses. Basically it's just Bernoulli percolation with an extra "loopless" condition. Let's assume for now that the infinite-volume measure is actually well-defined, unless this poses more than a technical problem (?).

Note that this model clearly does not satisfy the FKG inequality. Even worse, this model does not satisfy the finite-energy property, meaning that conditional percolation ratio on an edge $$e$$ (given arbitrary bond configurations on other edges $$E \setminus e$$) cannot be bounded away from zero (due to the loopless condition). There are several natural questions about this model that I cannot seem to settle with classical techniques for standard percolation models.

• Is there a phase transition for this model, in the sense that there exists $$p_c \in (0, 1)$$ where $$p_c = \inf\{ p: \text{there is at least one inifinite cluster with nonzero probability in } \phi_p \}$$. The Peierls contour argument doesn't seem to work due to the loopless condition. Neither does the classical self-dual argument in $$\mathbb{Z}^2$$ because I cannot find an obvious dual model. (Note $$p_c \geq 1/2$$ trivially because the model is stochastically dominated by the standard Bernoulli model.)
• If a phase transition does exist, in the (super)critical phase, is the number of infinite clusters almost surely a constant? If so, how many infinite clusters are there? And does this number change as $$p$$ is varied? Note that the Burton-Keane argument would not work (or needs to be modified) due to the lack of apparent ergodicity and finite-energy property.
• Regarding your second question, I have a forthcoming paper with Noah Halberstam where we prove that there is always at most one infinite cluster in three and four dimensions (in any translation invariant Gibbs measure for the model). Our arguments also give strong heuristic evidence that there should be infinitely many infinite clusters in the supercritical phase in five dimensions and higher (as there is in the USF, which is the p=1 limit).
– tmh
Sep 10, 2022 at 3:20
• @tmh that's very interesting! If I may ask, when you say "any translation invariant Gibbs measure", do you mean the two extremal ones induced by the free and wired b.c? Or perhaps there's some non-trivial multiplicity of extremal states for these models. Sep 10, 2022 at 7:18
• Currently we don't know anything about the structure of the Gibbs measures. It seems plausible that there is only one Gibbs measure in the supercritical regime, but the existing techniques don't really tell us anything AFAIK.
– tmh
Sep 10, 2022 at 8:34
• In particular I don't think we know that there is a well-defined "free Gibbs measure" or "wired Gibbs measure", just that translation invariant Gibbs measures always exist (by general abstract nonsense). For e.g. the random cluster model with q >= 1 the existence and extremality of the wired and free measures follows from FKG. The arboreal gas should probably be negatively correlated (which suggests that free and wired should be extremal) but this is not known.
– tmh
Sep 10, 2022 at 8:39
• It seems that the definition needs to be expanded because $\phi(\omega) = 0$ whenever $E$ is infinite. Sep 10, 2022 at 11:25

These fascinating questions have been studied recently, e.g. by Bauerschmidt, Crawford, Helmuth and Swan (no percolation on $$\mathbb{Z}^2$$) and by Bauerschmidt, Crawford and Helmuth (percolation phase transition on $$\mathbb{Z}^d$$ for $$d\geq 3$$).