Title: Loop vs. Bernoulli percolation on trees: strict inequality of critical values Show Chinese title

Title: 树上的环与伯努利渗流：临界值的严格不等式

Abstract: We consider loop ensembles on random trees. The loops are induced by a Poisson process of links sampled on the underlying tree interpreted as a metric graph. We allow two types of links, crosses and double bars. The crosses-only case corresponds to the random-interchange process, the inclusion of double bars is motivated by representations of models arising in mathematical physics. For a large class of random trees, including all Galton-Watson trees with mean offspring number in $(1,\infty)$, we show that the threshold Poisson intensity at which an infinite loop arises is strictly larger than the corresponding quantity for percolation of the untyped links. The latter model is equivalent to i.i.d. Bernoulli bond percolation. An important ingredient in our argument is a sensitivity result for the bond percolation threshold under downward perturbation of the underlying tree by a general finite range percolation. We also provide a partial converse to the strict-inequality result in the case of Galton-Watson trees and improve previously established criteria for the existence of an infinite loop in the case were the Galton-Watson tree has Poisson offspring. Show Chinese abstract

Abstract: 我们考虑随机树上的环集合。 环是由在底层树上采样的链接的泊松过程引起的，该树被解释为度量图。 我们允许两种类型的链接，交叉和双条。 仅包含交叉的情况对应于随机交换过程，引入双条是出于数学物理中出现的模型表示的动机。 对于一大类随机树，包括所有平均后代数在$(1,\infty)$中的 Galton-Watson 树，我们证明了出现无限环的阈值泊松强度严格大于无类型链接渗透的相应量。 后一种模型等价于独立同分布的伯努利边渗透。 我们论证中的一个重要组成部分是关于在一般有限范围渗透下对底层树进行向下扰动时边渗透阈值的敏感性结果。 我们还提供了在 Galton-Watson 树情况下严格不等式结果的部分逆定理，并改进了在 Galton-Watson 树具有泊松后代时存在无限环的先前建立的标准。