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- W2261587243 abstract "In this paper, we are interested in the loop cluster model on $mathbb{Z}^d$ for $dgeq 3$. 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 ($kappageq 0$) 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 $alpharightarrowkappa_c(alpha)$ for $alphageqalpha_c$, where $alpha_c$ is the critical value when $kappa=0$. In the end, we prove that $forallalpha>alpha_c$ 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 $alpha>alpha_c$." @default.
- W2261587243 created "2016-06-24" @default.
- W2261587243 creator A5058109823 @default.
- W2261587243 date "2015-04-29" @default.
- W2261587243 modified "2023-10-16" @default.
- W2261587243 title "Supercritical loop percolation on $mathbb{Z}^d$ for $dgeq 3$" @default.
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- W2261587243 doi "https://doi.org/10.48550/arxiv.1504.07906" @default.
- W2261587243 hasPublicationYear "2015" @default.
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