  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
$\alpha>\alpha_c$.
