Two-dimensional arboreal gas mass-gap conjecture

Let the arboreal gas be defined on $\mathbb{Z}^2$, let $\beta>0$, and write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree. Let $T_0$ be the tree containing the origin. Two-dimensional arboreal gas mass-gap conjecture. There exists $c_\beta>0$ such that

$$\mathbb{P}_{\beta}[i\leftrightarrow j]\approx e^{-c_\beta|i-j|}\qquad(i,j\in\mathbb{Z}^2).$$

As $\beta\to\infty$, the constant $c_\beta$ is exponentially small in $\beta$:

$$c_\beta\approx e^{-c\beta}.$$

In particular,

$$\mathbb{E}_{\beta}|T_0|\approx e^{c\beta}<\infty,$$

with a different constant $c$. This is a mass-gap-type conjecture motivated by the field-theoretic representation; the paper proves only almost-sure finiteness of all trees, not finite expected tree size, and the conjecture remains open.