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Circular law for non-Hermitian block band matrices with slowly growing bandwidth

Yi Han

TL;DR

This work proves the circular law for a class of non-Hermitian block band matrices with slowly growing bandwidth, showing that the empirical spectral distribution converges almost surely to the circular law whenever the block size $\ell_n$ tends to infinity and grows only polynomially with the dimension. The authors develop a novel transfer-operator framework on exterior algebra, couple it with a robust least-singular-value control under a bounded-density assumption, and employ a two-scale decomposition that reduces global determinant behavior to independent subsystems. A replacement principle links log determinants to the Ginibre potential, and a combination of rigidity estimates and matrix Dyson-equation analysis yields convergence of the Stieltjes transform and ultimately the ESD to $\mu_c$. The results extend circular-law universality into a localized band regime, providing the first rigorous confirmation in part of the localized spectrum and establishing a path toward understanding spectral statistics of structured non-Hermitian band matrices in regimes beyond delocalization.

Abstract

We consider the empirical eigenvalue distribution for a class of non-Hermitian random block tridiagonal matrices $T$ with independent entries. The matrix has $n$ blocks on the diagonal and each block has size $\ell_n$, so the whole matrix has size $n\ell_n$. We assume that the nonzero entries are i.i.d. with mean 0, variance 1 and having sufficiently high moments. We prove that when the entries have a bounded density, then whenever $\lim_{n\to\infty}\ell_n=\infty$ and $\ell_n=O(\operatorname{Poly}(n))$, the normalized empirical spectral distribution of $T$ converges almost surely to the circular law. The growing bandwidth condition $\lim _{n\to\infty}\ell_n=\infty$ is the optimal condition of circular law with small bandwidth. This confirms the folklore conjecture that the circular law holds whenever the bandwidth increases with the dimension, while all existing results for the circular law are only proven in the delocalized regime $\ell_n\gg n$.

Circular law for non-Hermitian block band matrices with slowly growing bandwidth

TL;DR

This work proves the circular law for a class of non-Hermitian block band matrices with slowly growing bandwidth, showing that the empirical spectral distribution converges almost surely to the circular law whenever the block size tends to infinity and grows only polynomially with the dimension. The authors develop a novel transfer-operator framework on exterior algebra, couple it with a robust least-singular-value control under a bounded-density assumption, and employ a two-scale decomposition that reduces global determinant behavior to independent subsystems. A replacement principle links log determinants to the Ginibre potential, and a combination of rigidity estimates and matrix Dyson-equation analysis yields convergence of the Stieltjes transform and ultimately the ESD to . The results extend circular-law universality into a localized band regime, providing the first rigorous confirmation in part of the localized spectrum and establishing a path toward understanding spectral statistics of structured non-Hermitian band matrices in regimes beyond delocalization.

Abstract

We consider the empirical eigenvalue distribution for a class of non-Hermitian random block tridiagonal matrices with independent entries. The matrix has blocks on the diagonal and each block has size , so the whole matrix has size . We assume that the nonzero entries are i.i.d. with mean 0, variance 1 and having sufficiently high moments. We prove that when the entries have a bounded density, then whenever and , the normalized empirical spectral distribution of converges almost surely to the circular law. The growing bandwidth condition is the optimal condition of circular law with small bandwidth. This confirms the folklore conjecture that the circular law holds whenever the bandwidth increases with the dimension, while all existing results for the circular law are only proven in the delocalized regime .

Paper Structure

This paper contains 22 sections, 32 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Models and main results
  3. A brief outline of the proof
  4. Proof of circular law: main structure of the argument
  5. Linear algebra results
  6. Least singular value and operator norm bounds
  7. Identification with a transfer operator iteration
  8. Exponential moments for intermediate increment steps
  9. Decomposition of the ergodic sum
  10. Least singular value for the auxiliary matrix
  11. Structure of normal vectors
  12. Invertibility via distance
  13. Back to the original matrix
  14. An upgraded tail estimate
  15. Convergence of the log determinant
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\zeta$ be a random variable satisfying that $\mathbb{E}[\zeta]=0,\mathbb{E}[|\zeta|^2]=1,$ and having all moments finite: for all $p\geq 2$, $\mathbb{E}[|\zeta|^{p}]<c_p<\infty$ for a $c_p>0$. We further assume that Take two integers $n,\ell\in\mathbb{N}$ and we denote $\ell=\ell_n$. Consider the following block tridiagonal matrix $T$ where each block $A_i,B_i,C_i$ has size $\ell\times\ell$

Theorems & Definitions (70)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 2.2
  • Corollary 2.4
  • proof : Proof of Proposition \ref{['proposition168']}
  • proof
  • ...and 60 more