Title: A characterization of strong percolation via disconnection Show Chinese title

Title: 通过断开的强渗透表征

Abstract: We consider a percolation model, the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$, $d \geq 3$, in the regime of parameters $u>0$ in which it is strongly percolative. By definition, such values of $u$ pinpoint a robust subset of the super-critical phase, with strong quantitative controls on large local clusters. In the present work, we give a new charaterization of this regime in terms of a single property, monotone in $u$, involving a disconnection estimate for $\mathcal{V}^u$. A key aspect is to exhibit a gluing property for large local clusters from this information alone, and a major challenge in this undertaking is the fact that the conditional law of $\mathcal{V}^u$ exhibits degeneracies. As one of the main novelties of this work, the gluing technique we develop to merge large clusters accounts for such effects. In particular, our methods do not rely on the widely assumed finite-energy property, which the set $\mathcal{V}^u$ does not possess. The charaterization we derive plays a decisive role in the proof of a lasting conjecture regarding the coincidence of various critical parameters naturally associated to $\mathcal{V}^u$ in a companion article. Show Chinese abstract

Abstract: 我们考虑一个渗流模型，即随机交错在$\mathbb{Z}^d$，$d \geq 3$上的空集$\mathcal{V}^u$，在参数范围$u>0$的情况下，它具有强渗流性。 根据定义，这样的$u$值指出了超临界相的一个稳健子集，并对大局部簇有强定量控制。 在本研究中，我们通过一个涉及$\mathcal{V}^u$的解连估计的单个性质，给出了该范围的新表征，该性质关于$u$是单调的。 一个关键方面是仅从这一信息本身展示大局部簇的粘合性质，而在此工作中的一大挑战是$\mathcal{V}^u$的条件分布表现出退化性。作为本工作的主要创新之一，我们开发的粘合技术用于合并大簇，考虑了这些效应。特别是，我们的方法不依赖于广泛假设的有限能量性质，而集合$\mathcal{V}^u$不具备该性质。我们推导出的特征在证明关于与$\mathcal{V}^u$自然相关的各种临界参数一致性的持久猜想的证明中起着决定性作用。