论文标题
稀疏随机简单复合物中的特征值和光谱差距
Eigenvalues and spectral gap in sparse random simplicial complexes
论文作者
论文摘要
我们认为，在$ n $ vertices上，邻接运算符$ a $ a $ a $ a $ a $ a $ a $ a $ x（d，n，p）$用于$ n $ dertices上的随机$ d $ d-d尺寸简单复合物，其中每个$ d $ cell in [0,1] $ in [0,1] $独立添加到完整的$（D-1）$ - （d-1）$ - skeleton。我们考虑稀疏的随机矩阵$ h $，它们是对中心和归一化的邻接矩阵$ \ MATHCAL {a}：=（np（1-p））^{ - 1/2} \ cdot（a- \ mathBb {e}任意有限分布$ z $。我们获得了预期的$ h $的预期界限，这使我们能够在特征值约束上证明结果，尤其是$ \ left \ welet \ vert h \ right \ right \ vert _ {2} $收敛到$ 2 \ sqrt {d} $ ncteration和$ \ \ mathbb { $ \ mathrm {var}（z）\ gg \ frac {\ log n} {n} $。证明中的主要成分是[lvhy18，定理4.8]对高维纯种复合物的上下文的概括，这可以被视为具有依赖条目的稀疏随机矩阵模型。
We consider the adjacency operator $A$ of the Linial-Meshulam model $X(d,n,p)$ for random $d-$dimensional simplicial complexes on $n$ vertices, where each $d-$cell is added independently with probability $p\in[0,1]$ to the complete $(d-1)$-skeleton. We consider sparse random matrices $H$, which are generalizations of the centered and normalized adjacency matrix $\mathcal{A}:=(np(1-p))^{-1/2}\cdot(A-\mathbb{E}\left[A\right])$, obtained by replacing the Bernoulli$(p)$ random variables used to construct $A$ with arbitrary bounded distribution $Z$. We obtain bounds on the expected Schatten norm of $H$, which allow us to prove results on eigenvalue confinement and in particular that $\left\Vert H\right\Vert _{2}$ converges to $2\sqrt{d}$ both in expectation and $\mathbb{P}-$almost surely as $n\to\infty$, provided that $\mathrm{Var}(Z)\gg\frac{\log n}{n}$. The main ingredient in the proof is a generalization of [LVHY18,Theorem 4.8] to the context of high-dimensional simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.
