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](https://arxiv.org/abs/0707.0073). 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](https://en.wikipedia.org/wiki/FKG_inequality). Even worse, this model does not satisfy the [finite-energy property](https://sites.math.northwestern.edu/%7Eauffing/papers/BurtonKeane.pdf), 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](https://sites.math.northwestern.edu/%7Eauffing/papers/BurtonKeane.pdf) argument would not work (or needs to be modified) due to the lack of apparent ergodicity and finite-energy property.



Infinite clusters for loopless percolation
These fascinating questions have been studied recently, e.g. by [Bauerschmidt, Crawford, Helmuth and Swan](https://arxiv.org/abs/1912.04854) (no percolation on $\mathbb{Z}^2$) and by [Bauerschmidt, Crawford and Helmuth](https://arxiv.org/abs/2107.01878) (percolation phase transition on $\mathbb{Z}^d$ for $d\geq 3$).


