CLT for generalized patterned random matrices: a unified approach

TL;DR

This paper develops a unified combinatorial moment-method framework to establish central limit theorems for linear eigenvalue statistics of generalized patterned random matrices with independent subexponential entries. For even degree monomial test functions, Gaussian LES is shown under a variance-growth condition $c^{p_n}N \le \mathrm{Var}(\mathrm{Tr}((A_n/\sqrt{N})^{p_n}))$, with $p_n=o(\log N/\log\log N)$, and fixed-$p$ results hold under stronger moment assumptions; odd-degree LES have limiting moments that depend on input moments and may be non-Gaussian unless moment-determinacy holds. The approach unifies results across Toeplitz, Hankel, circulant-type, and block-patterned matrices, providing structural conditions on link functions and circuits (via SP$(p,k)$) to guarantee normal LES, while offering detailed combinatorial bounds through generating vertices and Pi-sets. The framework delivers new LES results for classical and novel patterned models and extends to broader ensembles such as d-disco, checkerboard, swirl, and Hankel-type matrices, enriching the toolkit for spectral fluctuations in structured random matrices.

Abstract

In this paper, we derive a unified method for establishing the distributional convergence of linear eigenvalue statistics (LES) for generalized patterned random matrices. We prove that for an $N \times N$ generalized patterned random matrix with independent subexponential entries and even degree monomial test functions of degree $p_n=o(\log N/\log \log N)$, the LES converges to standard Gaussian distribution. This generalizes the CLT results on Gaussian patterned random matrices in Chatterjee(2009), Adhikari and Saha(2017). As an application, new results on LES of Toeplitz, Hankel, circulant-type matrices and block patterned random matrices for varying test functions are derived. For odd degree monomial test functions, we derive the limiting moments of LES and show that it may not converge to a Gaussian distribution.

Figures (6)

• Figure 1: A pictorial representation of the sentence $W=(w_1,w_2)$ obeying (\ref{['eqn: condition non-zero P24']})
• Figure 2: Example of a Type I subgraph with $a_e=\varphi(e)$ as distinct letters.
• Figure 3: (a) is an example of a sentence such that $R=0$ for all choice of $\zeta$ obeying \ref{['eqn: condition zeta']} and (b) is an example of a sentence such that $R=1$ for all choice of $\zeta$ obeying \ref{['eqn: condition zeta']}. In both the figures, for an edge $e$, the letter $\varphi(e)$ is given along the edge.
• Figure 4: Trees such that the distance of all vertices from the center is one. (a) shows the case when the center consists of only one vertex and (b) shows the case when the center consists of two vertices.
• Figure 5: (The construction of the subgraph $T_{W^\prime}$ from the graph $G_W$) The figure in the left represents the graph $G_W$. The center of $T_W$ is denoted by a larger red dot. The solid lines represent the edges of a spanning tree $T_W$ and the dotted lines the edges of $G_W$ not present in $T_W$. The figure in the right, $T_{W^\prime}$, is obtained from $T_W$ by removing the vertices at the distance two from the center of $T_W.$

Theorems & Definitions (54)

• Theorem 1
• Theorem 2
• Theorem 4
• Remark 5
• Theorem 6
• Remark 7
• Definition 8
• Definition 9
• Definition 10
• Definition 11