Supercritical long-range percolation on graphs of polynomial growth: the truncated one-arm exponent

We consider supercritical long-range percolation on transitive graphs of polynomial growth. In this model, any two vertices $x$ and $y$ of the underlying graph $G$ connect by a direct edge with probability $1-\exp(-\beta J(x,y))$, where $J(x,y)$ is a function that is invariant under the automorphism group of $G$, and we assume that $J$ decays polynomially with the graph distance between $x$ and $y$. We give up-to-constant bounds on the decay of the radius of finite cluster for $\beta > \beta_c$. In the same setting, we also give upper and lower bounds on the tail volume of finite clusters. The upper and lower bounds are of matching order, conjecturally on sharp volume bounds for spheres in transitive graphs of polynomial growth. As a corollary, we obtain a lower bound on the anchored isoperimetric dimension of the infinite component.