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The sparse circular law, revisited

Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney

TL;DR

This work resolves the sparse circular law for Bernoulli-type matrices in the regime $pn\to\infty$ by proving that the rescaled spectrum converges to the circular law for complex entries with unit variance, via a dynamically tracked singular-value evolution rather than $\varepsilon$-nets. The approach hinges on Girko's hermitization and a careful comparison between the full logarithmic potential $U_n(z)$ and a trimmed version $T_n(z)$ built from a principal minor, while incrementally exposing rows/columns and leveraging graph-structured quasi-randomness. Key innovations include unique-neighbourhood expansion, projection anti-concentration, and a stepwise process analysis ensuring drift that captures all new singular values; these yield convergence of the log-potential and hence of the empirical spectral measure to the circular law. The paper also extends the method to complex-valued $\xi$ with $\mathbb{E}|\xi|^2=1$, and links the tractable truncated potentials to Gaussian models to establish the limiting log-potential $U^{\circ}$. Overall, the results provide a streamlined, net-free framework for sparse random matrices and broaden the applicability of the circular law in the sparse regime.

Abstract

Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as $n\rightarrow \infty$ as long as $pn \rightarrow \infty$, which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when $pn$ is bounded. One feature of our proof is that it avoids the use of $ε$-nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.

The sparse circular law, revisited

TL;DR

This work resolves the sparse circular law for Bernoulli-type matrices in the regime by proving that the rescaled spectrum converges to the circular law for complex entries with unit variance, via a dynamically tracked singular-value evolution rather than -nets. The approach hinges on Girko's hermitization and a careful comparison between the full logarithmic potential and a trimmed version built from a principal minor, while incrementally exposing rows/columns and leveraging graph-structured quasi-randomness. Key innovations include unique-neighbourhood expansion, projection anti-concentration, and a stepwise process analysis ensuring drift that captures all new singular values; these yield convergence of the log-potential and hence of the empirical spectral measure to the circular law. The paper also extends the method to complex-valued with , and links the tractable truncated potentials to Gaussian models to establish the limiting log-potential . Overall, the results provide a streamlined, net-free framework for sparse random matrices and broaden the applicability of the circular law in the sparse regime.

Abstract

Let be an matrix with iid entries distributed as Bernoulli random variables with parameter . Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of is approximately uniform on the unit disk as as long as , which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when is bounded. One feature of our proof is that it avoids the use of -nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices as we incrementally expose the randomness in the matrix.
Paper Structure (13 sections, 26 theorems, 151 equations)

This paper contains 13 sections, 26 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Description of method
  3. A few preliminaries
  4. Unique neighbourhood expansions and quasi-randomness properties
  5. Spreadness of near-kernel vectors from graph quasi-randomness
  6. Projection anti-concentration
  7. Proof of the large h case
  8. Proof of the small h case
  9. A step in the process
  10. Analysis of the process
  11. Completion of the proof
  12. Proof of Lemma \ref{['lem:conv-Of-Tn']}: convergence of $T_n^{(1)}$, $T_n^{(2)}$
  13. Omitted proofs

Key Result

Theorem 1.1

Let $\xi$ be a complex random variable with mean $0$ and variance $1$. For each $n$, let $A_n$ be a random matrix with iid entries distributed as $\xi$. If we put $A^{\ast}_n = A_n \cdot n^{-1/2}$ then the spectral measure $\mu_{A_n^{\ast}}$ converges to the circular law in probability.

Theorems & Definitions (50)

  • Theorem 1.1: Tao and Vu
  • Theorem 1.2: Rudelson and Tikhomirov
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 5.1
  • proof
  • Lemma 5.3
  • ...and 40 more