论文标题
通过耦合的伯努利渗透的清晰度
Sharpness of Bernoulli percolation via couplings
论文作者
论文摘要
在本文中，我们考虑在本地有限的，及时的和无限的图上考虑Bernoulli Percolation（例如，高立方晶格$ \ Mathbb {Z}^d $）。我们证明了以下估计值，其中$θ_n（p）$是$ p $ open的路径从$ 0 $到半径$ n $：\ [\ [\ forall p \ in [0,1]，\ forall m，n \ ge 1，n \ ge 1，\quadθ_foral c \ frac {θ_n（p）} {2^{n/m}}。 \]这个结果意味着$θ_n（p）$在亚临界阶段呈指数衰减。这也意味着在超临界阶段的平均场下限。因此，我们提供了新的证据，证明了Bernoulli渗透的相转换的清晰度。与以前的清晰度证明相反，我们不依赖任何差异公式。主要的新颖性是由[Russo，1982]启发的随机统治结果。我们还讨论了在高维度中渗透结果的结果，在高维度上可以看作是近乎临界的清晰度估计。
In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $θ_n(p)$ is the probability that there is a path of $p$-open edges from $0$ to the sphere of radius $n$: \[ \forall p\in [0,1],\forall m,n \ge 1, \quad θ_{2n} (p-2θ_m(p))\le C\frac{θ_n(p)}{2^{n/m}}. \] This result implies that $θ_n(p)$ decays exponentially fast in the subcritical phase. It also implies the mean-field lower bound in the supercritical phase. We thus provide a new proof of the sharpness of the phase transition for Bernoulli percolation. Contrary to the previous proofs of sharpness, we do not rely on any differential formula. The main novelty is a stochastic domination result which is inspired by [Russo, 1982]. We also discuss a consequence of our result for percolation in high dimensions, where it can be seen as a near-critical sharpness estimate.
