arXiv subjects

Xiangyi Zhu

Publications and source records attributed to Xiangyi Zhu.

3 recordsLinked to original sources

arXiv2026

Generalized noise sensitivity of eigenvectors: All eigenvectors, inhomogeneous variance profiles, and dependent resampling

Chatterjee (2016) proved, as an application of his general framework relating superconcentration and chaos, that after the entries of an $n \times n$ matrix drawn from the Gaussian unitary ensemble undergo an entrywise Ornstein-Uhlenbeck (OU) process for time greater than $n^{-1/3}$, the top eigenvector of the matrix becomes almost completely decorrelated from its initial position. More recently, Bordenave, Lugosi, and Zhivotovskiy (2020) showed that the same happens under a discrete resampling model, once more than $n^{5/3}$ randomly chosen entries of a Wigner random matrix are resampled. We generalize these results in several directions: (1) we analyze the decorrelation of any eigenvector under continuous and discrete resampling dynamics, (2) we analyze the discrete resampling process for generalized Wigner matrices with inhomogeneous variance profiles, (3) we analyze a combination of continuous and discrete resampling where an OU process is repeatedly run for a certain time on randomly chosen entries, and (4) we analyze a dependent version of resampling where entries grouped into "blocks" of arbitrary shapes are resampled together. In each case, we show that a given eigenvector decorrelates provided that enough entries have been resampled or that the associated dynamics have been run for a long enough time. Our proofs take a different approach from prior work, relying more directly on the characterization of eigenvectors as derivatives of eigenvalues and reducing the problem of establishing eigenvector noise sensitivity to variants of standard and robust properties of random matrices such as bounds on eigenvalue spacings and eigenvector delocalization.

math.PR↗

arXiv2025

Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs across all sparsity regimes, providing explicit covariance formulas under the assumption that the variance profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the (non-centered) adjacency matrix and the centered adjacency matrix, with different scaling factors when the sparsity parameter $p$ satisfies $np = n^{Ω(1)}$, and with the same scaling factor when $np = n^{o(1)}$. In both cases, the limiting covariance is expressed in terms of homomorphism densities from certain types of finite graphs to a graphon. These results highlight a phase transition in the centering effect for global eigenvalue fluctuations. For the non-centered adjacency matrix, we also identify new phase transitions for the CLTs in the sparse regime when $n^{1/m} \ll np \ll n^{1/(m-1)}$ for $m \geq 2$. Furthermore, weaker conditions for the graphon convergence of the variance profile are sufficient as $p$ decreases from being constant to $np \to c\in (0,\infty)$. These findings reveal a novel connection between graphon limits and linear spectral statistics in random matrix theory.

math.PR↗

arXiv2024

Non-backtracking eigenvalues and eigenvectors of random regular graphs and hypergraphs

The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this paper, we take the first step in analyzing its bulk eigenvector behaviors. We demonstrate that for the non-backtracking operator $B$ of a random $d$-regular graph, its eigenvectors corresponding to nontrivial eigenvalues are completely delocalized with high probability. Additionally, we show complete delocalization for a reduced $2n \times 2n$ non-backtracking matrix $\tilde{B}$. By projecting all eigenvalues of $\tilde{B}$ onto the real line, we obtain an empirical measure that converges weakly in probability to the Kesten-McKay law for fixed $d\geq 3$ and to a semicircle law as $d \to\infty$ with $n \to\infty$. We extend our analysis to random regular hypergraphs, including the limiting measure of the real part of the spectrum for $\tilde{B}$, $\ell_{\infty}$-norm bounds for the eigenvectors of $\tilde{B}$ and $B$, and a deterministic relation between eigenvectors of $B$ and the eigenvectors of the adjacency matrix. As an application, we analyze the non-backtracking spectrum of the regular stochastic block model (RSBM) and provide a spectral method based on eigenvectors of $\tilde{B}$ to recover the community structure exactly. We also show that there exists an isolated real eigenvalue with an informative eigenvector inside the circle of radius $\sqrt{d_1+d_2-1}$ in the spectrum of $B$, analogous to the "eigenvalue insider" phenomenon for the Erdős-Rényi stochastic block model conjectured in Dall'Amico et al. (2019).

math.PR↗