Supercritical loop percolation on $\mathbb{Z}^d$ for $d\geq 3$

TL;DR

This paper studies a long-range loop percolation model on high-dimensional integer lattices, proving percolation in slabs, exponential decay of connectivity, and properties of the infinite cluster in the supercritical regime.

Contribution

It establishes percolation in truncated models above critical thresholds, analyzes the structure of the infinite cluster, and demonstrates the strict monotonicity of the critical curve with respect to parameters.

Findings

Percolation occurs in 2D slabs for large enough loops and parameters.

Exponential decay of one-arm connectivity in the supercritical regime.

Large balls in the infinite cluster are very regular, satisfying Harnack's inequality.

Abstract

In this paper, we are interested in the loop cluster model on $\mathbb{Z}^d$ for $d\geq 3$. It is a long range model with two parameters $\alpha$ and $\kappa$, where the non-negative parameter $\alpha$ measures the amount of loops, and $\kappa$ plays the role of killing on vertices penalizing ($\kappa\geq 0$) or favoring ($\kappa<0$) appearance of large loops. We consider the truncated loop cluster model formed by the Poisson point process $\mathcal{L}_{\alpha,\leq m}$, which is the restriction of $\mathcal{L}_{\alpha}$ on loops with at most $m$ jumps. We prove the existence of percolation in a $2$-dimensional slab for the truncated loop model $\mathcal{L}_{\alpha,\leq m}$ as long as the intensity parameter $\alpha$ is strictly above the critical threshold of the non-truncated loop model and $m$ is large enough. We apply this result to prove the exponential decay of one arm connectivity for the finite cluster at $0$ for the whole supercritical regime of the non-truncated loop model. For $\kappa=0$, this loop percolation model provides an example in which we have different behaviors of finite clusters in sub-critical and super-critical regimes. Also, we deduce the strict increase of the critical curve $\alpha\rightarrow\kappa_c(\alpha)$ for $\alpha\geq\alpha_c$, where $\alpha_c$ is the critical value when $\kappa=0$. In the end, we prove that $\forall\alpha>\alpha_c$ large balls in the infinite cluster are finally very regular in the sense of \cite{Sapozhnikov2014}, which implies that large balls are finally very good in the sense of \cite{BarlowMR2094438}. By \cite{BarlowMR2094438} and \cite{BarlowHamblyMR2471657}, we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all $\alpha>\alpha_c$.