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Rank of Sparse Bernoulli Matrices

Han Huang

TL;DR

This paper resolves the rank/invertibility behavior of sparse Bernoulli matrices $A_n$ in the regime $\frac{\log n}{n}\le p\le c_{\beta}$, showing that the probability of a small least singular value is governed by local obstructions (zero rows/columns) rather than global structure. The core method blends a refined vector-decomposition into steep ($\mathcal{T}$), robust ($\mathcal{R}$), and gradual ($\mathcal{V}$) classes with a targeted submatrix reduction $A_{I,J}$ and the Litvak–Tikhomirov invertibility via distance framework, augmented by Rogozin anticoncentration and expansion arguments. A key novelty is the treatment of steep vectors through a phase-transition analysis around the critical window $p \asymp \tfrac{\log n}{n}$, including a carefully chosen threshold $s_0$ and a mechanism to ensure a favorable $(I,J)$ submatrix exists with high probability. The main result expresses the precise small-singular-value probability as a sum of an $t$-term and a local-zero-row/column probability, extending LT20 and yielding a sparse-regime counterpart of the strong singularity conjecture for Bernoulli matrices, with application to rank questions and a refined understanding of how local obstructions dominate invertibility. The techniques have broad relevance for random discrete matrices, combining anti-concentration, expansion, and structured-vector partitions to obtain sharp probabilistic invertibility results in the sparse-to-dense transition.

Abstract

Let $ A_n $ be an $n \times n$ random matrix with i.i.d Bernoulli($p$) entries. For a fixed positive integer $β$, suppose $p$ satisfies $$ \frac{ \log(n) }{ n } \le p \le c_β$$ where $c_β\in ( 0, 1/2 )$ is a $β$-dependentvalue. For $t \ge 0$, $$ \mathbb{P} \left\{ s_{ n - β+ 1}(A) \le t n^{-2β+ \mathfrak{n}(1) }(pn)^{-7} \right\} = t + ( 1 + o_\mathfrak{n}(1) ) \mathbb{P} \bigg\{ \mbox{either $β$ rows or $β$ columns of $A_n$ equal $\vec{0}$} \bigg\}. $$

Rank of Sparse Bernoulli Matrices

TL;DR

This paper resolves the rank/invertibility behavior of sparse Bernoulli matrices in the regime , showing that the probability of a small least singular value is governed by local obstructions (zero rows/columns) rather than global structure. The core method blends a refined vector-decomposition into steep (), robust (), and gradual () classes with a targeted submatrix reduction and the Litvak–Tikhomirov invertibility via distance framework, augmented by Rogozin anticoncentration and expansion arguments. A key novelty is the treatment of steep vectors through a phase-transition analysis around the critical window , including a carefully chosen threshold and a mechanism to ensure a favorable submatrix exists with high probability. The main result expresses the precise small-singular-value probability as a sum of an -term and a local-zero-row/column probability, extending LT20 and yielding a sparse-regime counterpart of the strong singularity conjecture for Bernoulli matrices, with application to rank questions and a refined understanding of how local obstructions dominate invertibility. The techniques have broad relevance for random discrete matrices, combining anti-concentration, expansion, and structured-vector partitions to obtain sharp probabilistic invertibility results in the sparse-to-dense transition.

Abstract

Let be an random matrix with i.i.d Bernoulli() entries. For a fixed positive integer , suppose satisfies where is a -dependentvalue. For ,

Paper Structure

This paper contains 34 sections, 49 theorems, 501 equations.

Table of Contents

  1. Introduction
  2. Acknowlegement
  3. Outline of proof
  4. Choice of the parameter $\boldsymbol{s_0}$.
  5. Notations and standard Probability Estimates
  6. Treatment for Steep Vectors
  7. Statistics of Sparse Rows and Columns and Expansion Property of $A$
  8. Estimate of $|\mathcal{J}(s)|$ and $\mathbb{P}\{ \Omega_{\rm RC}^c(\beta)\}$
  9. Expansion Property
  10. Typical Realization of $A$ for $\frac{ \log( n ) }{ n } \le p \le ( 1 + \frac{ 1 }{ 2 \beta } ) \frac{ \log( n ) }{ n }$
  11. Probability Estimate for $\Omega_{\ref{['thm: SteepExpansion']} }^c$.
  12. Proof of Lemma \ref{['lem: Omega_W']}
  13. Proof of Theorem \ref{['thm: SteepExpansion']} and Theorem \ref{['thm: T_1s']}
  14. Typical $A$ for $p \ge ( 1 + \frac{ 1 }{ 2\beta } )\frac{ \log( n ) }{ n }$.
  15. Probability Estimate for $\Omega_{\ref{['thm: SteepExpansion tau>10']}}^c$.
  16. ...and 19 more sections

Key Result

Theorem 1.2

There is a universal constant $C>1$ with the following property. Let $A_n$ be an $n \times n$ random matrix whose entries are i.i.d Bernoulli($p$), with $p = p_n$ satisfying Then, when $n$ is sufficiently large, for $t \ge 0$, Further, this estimate can be improved when $p$ is also bounded below by a constant. Let $q \in (0, C^{-1})$ be a parameter. Then, there exists $C_q >0$ such that if $p \g

Theorems & Definitions (99)

  • Conjecture 1.1: Strong singularity conjecture for Bernoulli matrices
  • Theorem 1.2: Litvak-Tikhomirov LT20
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • ...and 89 more