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

TL;DR

The paper analyzes supercritical long-range percolation on transitive graphs of polynomial growth with a transitive, polynomially decaying kernel $J(x,y) = \Theta(d_G(x,y)^{-d\alpha})$ for $\alpha>1$. It develops a block-decomposition framework, leverages Timár’s coarse connectivity and mass-transport, and applies $d$-dimensional and Varopoulos-type isoperimetric inequalities to bound long-range connections; a pigeonhole argument translates truncated one-arm events into long-edge contributions, enabling sharp decay bounds for finite-cluster radii and both strong and weak tail bounds for finite-cluster sizes in different $\alpha$-regimes. A key outcome is that, for $\alpha>1+1/d$, finite-cluster tails decay as $\exp(-\beta k^{(d-1)/d}/A)$, while for $\alpha\in(1,2)$ the weak-decay regime yields $\exp(-k^{2-\alpha}/A)$ tails, with a transition at $\alpha=1+1/d$ marking a shift to surface-tension-dominated behavior. The results imply an anchored isoperimetric lower bound for the infinite cluster and, under sphere-volume assumptions, provide matching lower bounds up to polylogarithmic corrections, shedding light on the geometric structure of the supercritical long-range percolation cluster and extending sharp percolation results from $\mathbb{Z}^d$ to transitive graphs of polynomial growth.

Abstract

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 $1-\exp(-βJ(x,y))$, 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 $β> β_c$. 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.

Figures (3)

• Figure 1: The results of \ref{['thm:cluster_size_strong_decay', 'thm:cluster_lower_bound', 'thm:cluster_size_weak_decay']} as a function of the long-range parameter $\alpha$, conjectural on sharp bounds for the volume of spheres in transitive graphs of polynomial growth. We prove both upper and lower bounds for all $\beta>\beta_c$ when $\alpha\in(1,2)$ in moreno-alonso_supercritical_2025. Here, $d \geq 2$.
• Figure 2: Figures \ref{['fig:lrp_configuration']} and \ref{['fig:associated_decomposition']} show a sketch of a long-range percolation configuration and the associated block decomposition. In Figure \ref{['fig:isolation']} the red path is a coarsely connected path which starts from a boundary vertex and whose subsequent vertices are all guaranteed to not be in $K$, so that all edges from the boundary vertex to path vertices must be closed. This will be used to bound the isolation of blocks.
• Figure 3: Figures \ref{['fig:block_and_spanning_graph']} and \ref{['fig:weighted_block_graph']} represent the block graph and the weighted block graph associated to the percolation configuration in Figure \ref{['fig:lrp_configuration']} with an arbitrary choice of labeling. In Figure \ref{['fig:block_and_spanning_graph']}, the highlighted subgraph is a spanning tree of the block graph, and the vector of forward degrees $\mathbf{f} = (2,1,1,0,2,0,0)$ is an $\mathbf{f}$-tree.

Theorems & Definitions (73)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3: Cluster-size decay
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9: Cluster-size decay in the weak-decay regime moreno-alonso_supercritical_2025
• Lemma 1.10