Inside the critical window for cohomology of random k‐complexes
Inside the critical window for cohomology of random k‐complexes
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随机 k 复合体上同调的临界窗口内
DOI：
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发表时间：
2013
期刊：
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通讯作者：
B. Pittel
中科院分区：
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作者：
Matthew Kahle;B. Pittel
我们证明了Linial-Meshulam和Meshulam-Wallach定理的更清晰的版本，这些定理描述了随机k维单纯复形在窄过渡窗口内的（λ/2）-上同调行为。特别地，我们证明了，如果Y是一个随机k维单纯复形，每个k-单纯形出现i.i.d.若概率p=klogn+cn，k≥1且c∈ k固定，则上同调βk−1（Y）的维数是渐近泊松分布，平均为e−c/k！.在k = 2的情况下，我们还证明了在伴随的增长过程中，Hk−1（Y，λ/2）以高概率恰好在最后一个（k−1）-单纯形被k-单纯形覆盖的时刻消失，这是Bollobás和Bollason关于随机图连通性的“停时”定理的高维模拟。随机结构算法，2015 © 2015 Wiley Periodicals，Inc.随机结构算法，48，102-124，2016
We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for (ℤ/2) ‐cohomology of a random k‐dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k‐dimensional simplicial complex with each k‐simplex appearing i.i.d. with probability p=klogn+cn, with k≥1 and c∈ℝ fixed, then the dimension of cohomology βk−1(Y) is asymptotically Poisson distributed with mean e−c/k! . In the k = 2 case we also prove that in an accompanying growth process, with high probability, Hk−1(Y,ℤ/2) vanishes exactly at the moment when the last (k−1) ‐simplex gets covered by a k‐simplex, a higher‐dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016