Eigenvalue statistics of block-Wigner-type matrices and unfolding matrices of supersymmetric tensors

Abstract

Random matrix theory has been very active and popular in high-dimensional statistics and data analysis for the last two decades. It has been providing some fundamental tool-sets for studying the eigenvalues and eigenvector statistics of various matrix models in a wide variety of topics.

In the meantime, there have been growing interests in modeling social networks and other complex networks among researchers with various backgrounds. Classic probabilistic models such as the Erd\H{o}s-R\'enyi model, stochastic block model, Chung-Lu models have been heavily studied. In this thesis, the major motivation is to study the spectral properties of the stochastic block model via the random matrices approach. To be more precise, the central objects of study in this thesis are the linear eigenvalue statistics of the renormalized adjacency matrices of the random graphs. Namely, many random network models can be identified as undirected random graphs $(E,V)$ which have independent edges with connection probabilities assigned to be $P(\{i, j\} \in E) = p_{ij}$, i.e., the adjacency matrix of such graph would be a symmetric matrix with independent entries $A_{ij} \sim Ber(p_{ij})$ on the upper triangular part, where $p_{ij}$'s follow certain patterns. The patterns corresponding to the stochastic block models will lead to the Wigner-type matrices with a block structure. In general, the investigation into the eigenvalue and eigenvector statistics can help to determine the patterns, estimate related parameters, or construct hypothesis testing about these parameters. Such applications serve as a strong motivation for studying the linear spectral statistics of such adjacency matrices from random graphs after proper renormalizations.

In the first part of this thesis, we study the first renormalization of the adjacency matrices, which would be Wigner matrices with however inhomogeneous fourth moments, and establish a CLT for the linear spectral statistics of such matrices.

Second, we investigate another type of renormalization that leads to a new class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics via the resolvent method. Further, we show that for the certain estimator of the renormalized adjacency matrices, which will be no longer be Wigner-type but share long-range non-decaying weak correlations among the entries, the linear spectral statistics will still follow the same law as the block-Wigner-type matrices, thus enabling hypothesis testing about stochastic block model.

Third, this thesis proposes a limiting spectral distribution result for the unfolding matrices of a special class of random tensors, namely the supersymmetric tensor with i.i.d. entries up to the permutation invariance, we show that under certain conditions, the limiting spectral distribution of such unfolding matrices would be a mixture of a point mass at the origin with a scaled semicircle law.