随机0/1多面体迅速扩展
Random 0/1-polytopes expand rapidly
AI总结：
本文研究了随机0/1多面体的边扩展性，证明当顶点以常概率采样时，多面体的边扩展性显著增强，分别在p>1/2和p<1/2时达到Θ(n)和n^{Θ(log log n)}的下界，改进了之前的结果。
摘要：
一个0/1多面体是{0,1}^n的一个子集V的凸包。Mihail和Vazirani的著名猜想指出，每个0/1多面体的图的边扩展性至少为1。本文证明，典型的0/1多面体具有更强的扩展性。具体而言，若V由独立采样{0,1}^n的每个顶点以常概率p形成，则当p∈(1/2,1)时，边扩展性以高概率为Θ(n)，当p∈(0,1/2)时为n^{Θ(log log n)}。这改进了Ferber、Krivelevich、Sales和Samotij之前的最佳下界Ω(1)。
英文摘要：
A 0/1-polytope is the convex hull of a subset $V\subseteq \{0,1\}^n$. A celebrated conjecture of Mihail and Vazirani asserts that the graph of every 0/1-polytope has edge-expansion at least 1. In this paper, we show that typical 0/1-polytopes have significantly stronger expansion. Specifically, if $V$ is formed by sampling each vertex of $\{0,1\}^n$ independently with constant probability $p$, then with high probability the edge-expansion is $Θ(n)$ for $p \in (1/2, 1)$, and $n^{Θ(\log \log n)}$ for $p \in (0, 1/2)$. This improves the previously best known bound $Ω(1)$ due to Ferber, Krivelevich, Sales and Samotij.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
