The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex \(Y_2(n,p)\) was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of \(Y_2(n,p)\) is about \(p\approx n^{-1/2}\). In this paper, we show that this threshold probability is at most \(p\le (\gamma n)^{-1/2}\), where \(\gamma =4^4/3^3\), and conjecture that this threshold is sharp. In fact, we show that \(p=(\gamma n)^{-1/2}\) is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of \(Y_2(n,p)\) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.

Babson、Hoffman 和 Kahle 首次研究了二维 Linial-Meshulam 随机单纯复形\(Y_2(n,p)\)的基本群。他们证明了\(Y_2(n,p)\) 的简单连通性的阈值概率约为\(p\approx n^{-1/2}\)。在本文中，我们表明该阈值概率至多为\(p\le (\gamma n)^{-1/2}\)，其中\(\gamma =4^4/3^3\)，以及推测这个阈值是尖锐的。事实上，我们证明\(p=(\gamma n)^{-1/2}\)是更强属性的一个尖锐的阈值概率，即每个长度为 3 的循环都是\(Y_2( n,p)\)同胚于磁盘。我们的证明使用泊松范式，并依赖于 Tutte 对平面三角剖分的枚举的经典结果。