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).
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Andrew Newman (2024) studied this question.
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