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

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 $\alpha>\alpha_c$.