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arXiv 1504.07906math.PR
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$d\geq 3$在$\mathbb{Z}^d$上的超临界循环渗流

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

Yinshan Chang

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摘要：

在本文中，我们感兴趣的是在$\mathbb{Z}^d$ $d\geq 3$上的环簇模型。它是一个具有两个参数$α$和$κ$的长程模型，其中非负参数$α$测量循环的数量，$κ$起着杀死顶点惩罚（$κ\geq 0$）或支持（$κ<0$）大循环出现的作用。考虑由Poisson点过程$\mathcal{L}_{α,\leq m}$构成的截断环簇模型，它是$\mathcal{L}_α$对最多有$m$跳的环的限制.本文证明了在$2$维平板中，对于截断环模型$\mathcal{L}_{α,\leq m}$，只要强度参数$α$严格大于非截断环模型的临界阈值，且$m$足够大，就存在渗流.我们应用这个结果来证明在$0$为整个超临界制度的非截断环模型的有限集群的单臂连接的指数衰减。对于$κ=0$，这个回路渗流模型提供了一个例子，在这个例子中，我们有不同的行为，在亚临界和超临界制度的有限团簇。此外，我们推导出严格增加的临界曲线$α\rightarrowκ_c(α)$的$α\geqα_c$，其中$α_c$是当$κ=0$的临界值。最后，我们证明了无限簇中的$\forallα>α_c$大球在\cite{Sapozhnikov2014}意义下最终是非常正则的，这意味着大球在\cite{BarlowMR2094438}意义下最终是非常好的.利用\cite{BarlowMR2094438}和\cite{BarlowHamblyMR2471657}，对所有的$α>α_c$，我们得到了无限簇上简单随机游动的Harnack不等式和Gaussian型估计.

英文摘要：

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 $α$ and $κ$, where the non-negative parameter $α$ measures the amount of loops, and $κ$ plays the role of killing on vertices penalizing ($κ\geq 0$) or favoring ($κ<0$) appearance of large loops. We consider the truncated loop cluster model formed by the Poisson point process $\mathcal{L}_{α,\leq m}$, which is the restriction of $\mathcal{L}_α$ on loops with at most $m$ jumps. We prove the existence of percolation in a $2$-dimensional slab for the truncated loop model $\mathcal{L}_{α,\leq m}$ as long as the intensity parameter $α$ 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 $κ=0$, 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 $α\rightarrowκ_c(α)$ for $α\geqα_c$, where $α_c$ is the critical value when $κ=0$. In the end, we prove that $\forallα>α_c$ 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 $α>α_c$.

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