论文标题
稀疏随机矩阵的光谱大偏差
Spectral large deviations of sparse random matrices
论文作者
论文摘要
Wigner矩阵的特征值一直是调查的主要主题。此类随机矩阵的一个特别重要的子类是由配备I.I.D.的Erdős-rényi图$ \ Mathcal {G} _ {n，p} $的邻接矩阵形成的。边缘重量。可观察到的特别是最大的特征值。在本文中，我们研究了此类矩阵最大的特征值的巨大偏差行为，多年来，这个主题受到了很大的关注。我们专注于$ p = \ frac {d} {n} $，其中大多数已知的技术分解。到目前为止，结果仅以$ \ Mathcal {g} _ {n，\ frac {d} {n}} $而没有边缘赋予（Krivelevich and Sudakov，'03），（Bhattacharya，Bhattacharya，Bhattacharya，and angulya＆danguly，'21），以及Gaussian Edge-Edge-weighs（bhattacharya）（bhattacharya）（ 在本文中，我们考虑了一般体重分布的效果。更具体地说，我们考虑的条目的条目以$ e^{ - t^α} $的尾巴衰减，带有$α> 0 $，其中$ 0 <α<2 $和$α> 2 $的条目分别比高斯式尾巴更重，更轻。 While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when $α> 2.$ In contrast, in the $α< 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin-Straus theorem, a classical result from spectral graph theory. 作为我们大偏差结果的副产品，我们还为最大的特征值建立了大量结果的新定律。特别是，我们表明，最大特征值的典型值在$α= 2 $，即高斯分布时表现出相变。
Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices is formed by the adjacency matrix of an Erdős-Rényi graph $\mathcal{G}_{n,p}$ equipped with i.i.d. edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. We focus on the case $p = \frac{d}{n}$, where most known techniques break down. So far, results were known only for $\mathcal{G}_{n,\frac{d}{n}}$ without edge-weights (Krivelevich and Sudakov, '03), (Bhattacharya, Bhattacharya, and Ganguly, '21) and with Gaussian edge-weights (Ganguly and Nam, '21). In the present article, we consider the effect of general weight distributions. More specifically, we consider the entries whose tail probabilities decay at rate $e^{-t^α}$ with $α>0$, where the regimes $0<α<2$ and $α>2$ correspond to tails heavier and lighter than the Gaussian tail respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when $α> 2.$ In contrast, in the $α< 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin-Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish new law of large numbers results for the largest eigenvalue. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at $α= 2$, i.e. the Gaussian distribution.
