Unimodular finite-size criterion for subexponential connection decay

Let $\mathcal U_d^*$ be the class of infinite unimodular transitive graphs of degree $d$ that are not one-dimensional. For a graph $G$ with origin $o$, let $B_n$ be the radius-$n$ ball around $o$, let $S_m$ be the sphere of radius $m$, and write $\mathbb P_p(u\leftrightarrow v)$ for the Bernoulli bond percolation connection probability. Unimodular reformulation of locality. For all $d\in\mathbb N$ and all $\varepsilon,\delta>0$, there exists $n\in\mathbb N$ such that, for every $G\in\mathcal U_d^*$ and $p\in[0,1]$,

$$\min_{u,v\in B_n}\mathbb P_p(u\leftrightarrow v)\geq\varepsilon\quad\Longrightarrow\quad\lim_{m\to\infty}\frac1m\log\mathbb P_{p+\delta}(o\leftrightarrow S_m)=0.$$

This is presented as a further reformulation in the unimodular case, using sharpness of the phase transition. It is part of the paper's reformulation framework for the locality conjecture and remains conjectural in the stated generality.