关于随机二部双正则图的第二特征值
On the second eigenvalue of random bipartite biregular graphs
AI总结：
本文研究随机二部双正则图邻接矩阵的第二特征值，利用尺寸偏置耦合和新切换操作证明其谱隙为 $O(\sqrt{d_1})$，并推广到随机正则有向图的第二奇异值估计。
摘要：
我们考虑均匀选取的随机 $(d_1,d_2)$-双正则二部图 $G$ 的谱隙，其中 $|V_1|=n, |V_2|=m$，且 $d_1,d_2$ 可能随 $n$ 和 $m$ 增长。设 $A$ 为 $G$ 的邻接矩阵。在假设 $d_1\geq d_2$ 且 $d_2=O(n^{2/3})$ 的条件下，我们证明以高概率有 $λ_2(A)=O(\sqrt{d_1})$。作为推论，结合 Tikhomirov 和 Youssef (2019) 的结果，我们证明了均匀随机 $d$-正则有向图的第二奇异值在 $1\leq d\leq n/2$ 时以高概率为 $O(\sqrt{d})$。进一步假设 $d_2$ 固定且 $d_1=O(n^2)$，我们证明对于随机 $(d_1,d_2)$-双正则二部图，以高概率对所有 $2\leq i\leq n+m-1$ 都有 $|λ_i^2(A)-d_1|=O(\sqrt{d_1})$。这两个结果的证明基于 Cook、Goldstein 和 Johnson (2018) 为随机 $d$-正则图引入的尺寸偏置耦合方法，以及我们为随机二部双正则图定义的若干新的切换操作。
英文摘要：
We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption that $d_1\geq d_2$ and $d_2=O(n^{2/3}),$ we show that $λ_2(A)=O(\sqrt{d_1})$ with high probability. As a corollary, combining the results from Tikhomirov and Youssef (2019), we showed that the second singular value of a uniform random $d$-regular digraph is $O(\sqrt{d})$ for $1\leq d\leq n/2$ with high probability. Assuming $d_2$ is fixed and $d_1=O(n^2)$, we further prove that for a random $(d_1,d_2)$-biregular bipartite graph, $|λ_i^2(A)-d_1|=O(\sqrt{d_1})$ for all $2\leq i\leq n+m-1$ with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook, Goldstein, and Johnson (2018) for random $d$-regular graphs and several new switching operations we defined for random bipartite biregular graphs.
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