随机升力谱
The spectrum of random lifts
摘要：
对于固定d-正则图H，通过用一个包含n个顶点的“纤维”代替H的每个顶点v，然后在对应于H的邻顶点的纤维之间放置一个一致随机匹配，得到一个随机n-提升.我们表明，以极高的概率，所有的电梯，不是H的本征值的本征值，有秩序O（sqrt（d））。特别地，如果H是Ramanujan，那么它的n-提升有很高的概率接近Ramanujan。我们还表明，任何异常大的本征值的n-电梯，而不是本征值的H，是压倒性的可能已造成的一个稠密的子图的大小为O（|E（H）|).
英文摘要：
For a fixed d-regular graph H, a random n-lift is obtained by replacing each vertex v of H by a "fibre" containing n vertices, then placing a uniformly random matching between fibres corresponding to adjacent vertices of H. We show that with extremely high probability, all eigenvalues of the lift that are not eigenvalues of H, have order O(sqrt(d)). In particular, if H is Ramanujan then its n-lift is with high probability nearly Ramanujan. We also show that any exceptionally large eigenvalues of the n-lift that are not eigenvalues of H, are overwhelmingly likely to have been caused by a dense subgraph of size O(|E(H)|).
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。
