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.
Best Answer
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 $d\geq 3$).