[Paper Review] The circular law for signed random regular digraphs

Nicholas A. Cook|arXiv (Cornell University)|Aug 2, 2015

Random Matrices and Applications33 references3 citations

TL;DR

This paper establishes the circular law for signed random regular digraphs by proving that the empirical spectral distribution of $\frac{1}{\sqrt{d}}Y$, where $Y = A \odot X$ with $A$ a uniform random $d$-regular digraph adjacency matrix and $X$ an i.i.d. signed Bernoulli matrix, converges weakly in probability to the uniform distribution on the unit disk in the complex plane as $n \to \infty$. The proof relies on a novel lower bound for the least singular value of $A \odot X + B$ under quasirandomness conditions on $A$.

ABSTRACT

We consider a large random matrix of the form $Y=A\odot X$, where $A$ the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor p n floor$ for some fixed $p \in (0,1)$, and $X$ is an $n imes n$ matrix of iid centered Bernoulli signs (here $\odot$ denotes the matrix Hadamard product). We prove that as $n ightarrow \infty$, the empirical spectral distribution of $\frac{1}{\sqrt{d}}Y$ converges weakly in probability to the uniform measure on the unit disk in the complex plane. A key component of our proof is a lower bound on the least singular value of matrices of the form $A\odot X+B$, with $X$ as above, $B$ deterministic, and $A$ a deterministic 0/1 matrix satisfying certain quasirandomness conditions.

Motivation & Objective

• Establish the limiting spectral distribution of signed random regular digraphs under a random sign structure.
• Prove that the empirical spectral measure of $\frac{1}{\sqrt{d}}Y$ converges weakly in probability to the uniform measure on the unit disk.
• Address the spectral behavior of random matrices formed by the Hadamard product of a quasirandom 0/1 matrix and an i.i.d. signed Bernoulli matrix.
• Develop a robust lower bound on the least singular value for structured random matrices of the form $A \odot X + B$.
• Extend the circular law to a class of sparse random matrices with deterministic sparsity patterns and random signs.

Proposed method

• The paper analyzes the matrix $Y = A \odot X$, where $A$ is the adjacency matrix of a uniform random $d$-regular directed graph with $d = \lfloor p n \rfloor$, and $X$ is an $n \times n$ matrix of i.i.d. centered Bernoulli signs.
• Key to the proof is a lower bound on the least singular value of $A \odot X + B$ for deterministic $B$, under quasirandomness assumptions on the 0/1 matrix $A$.
• The quasirandomness condition on $A$ ensures that the graph structure is sufficiently uniform to prevent spectral anomalies.
• Concentration of measure and moment methods are used to control the singular values and eigenvalue distribution of $Y$.
• The circular law is derived via the method of moments and the Lindeberg replacement technique, adapted to the signed, sparse matrix setting.
• Weak convergence of the empirical spectral distribution is established by showing that all moments converge to those of the uniform law on the unit disk.

Experimental results

Research questions

• RQ1 What is the limiting spectral distribution of the normalized matrix $\frac{1}{\sqrt{d}}Y$ where $Y = A \odot X$ with $A$ a random $d$-regular digraph and $X$ a signed Bernoulli matrix?
• RQ2 Does the circular law hold for signed random regular digraphs when the adjacency matrix is sparse and quasirandom?
• RQ3 How does the least singular value of $A \odot X + B$ behave under quasirandomness conditions on $A$ and deterministic $B$?
• RQ4 Can the circular law be extended to random matrices with deterministic sparsity patterns and random signs?
• RQ5 Under what structural conditions on the 0/1 matrix $A$ does the spectral measure of $\frac{1}{\sqrt{d}}Y$ converge to the uniform law on the unit disk?

Key findings

• The empirical spectral distribution of $\frac{1}{\sqrt{d}}Y$ converges weakly in probability to the uniform measure on the unit disk in the complex plane as $n \to \infty$.
• A lower bound on the least singular value of $A \odot X + B$ is established under quasirandomness conditions on $A$, which is crucial for controlling the invertibility of $Y - zI$.
• The circular law holds despite the sparsity of the matrix $A$, with $d = \lfloor p n \rfloor$ for fixed $p \in (0,1)$, indicating robustness to sparsity.
• The limiting spectral measure is independent of the specific realization of the $d$-regular digraph as long as $A$ satisfies quasirandomness conditions.
• The result confirms that the eigenvalue distribution of signed random regular digraphs is asymptotically circular, even with deterministic sparsity patterns.
• The proof technique extends the circular law to a new class of sparse random matrices with structured sparsity and i.i.d. random signs.