We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result, we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so, we show that there is an intermediate probability regime in which the random flag complex has fundamental group that is neither free nor has Kazhdan's property (T).

中文翻译：

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随机标志复合体同源性的单边尖锐阈值

我们证明随机标志复数具有一个概率状态，其中非零同源性的概率渐近限制在远离零和远离一的范围内。与这个主要结果相关，我们还在尖锐阈值上建立了新的界限，使随机标志复合体的基本群成为自由群。在此过程中，我们表明存在一个中间概率状态，其中随机标志复合体具有既不自由也不具有 Kazhdan 属性 (T) 的基本群。