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On the singularity probability of discrete random matrices

Jean Bourgain, Van Vu, Philip Matchett Wood

TL;DR

This work advances the understanding of singularity probabilities for discrete random matrices by proving a general exponential upper bound under a $p$-boundedness framework and a structured value-set. The approach combines a finite-field reduction with a Structure Theorem akin to Freiman-type results, and employs Halász-type Fourier analysis to partition hyperplanes by combinatorial dimension into manageable cases. Consequently, it yields improved bounds, notably $\Pr(M_{\pm 1,n}\text{ is singular}) \le (1/\sqrt{2}+o(1))^n$ for Bernoulli entries, and extends to matrices with complex entries, non-identical distributions, and partially random structures, including results on rational eigenvalues for integer matrices. The methods provide a robust, broadly applicable toolkit for exponential bounds in discrete random matrix settings, with potential impact on random combinatorics and numerical linear algebra.

Abstract

Let $M_n$ be an $n$ by $n$ random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that $M_n$ is singular is at most $(1/\sqrt{2} + o(1))^n$, improving on the previous best upper bound of $(3/4 + o(1))^n$ proven by Tao and Vu in arXiv:math/0501313v2. This paper follows a similar approach to the Tao and Vu result, including using a variant of their structure theorem. We also extend this type of exponential upper bound on the probability that a random matrix is singular to a large class of discrete random matrices taking values in the complex numbers, where the entries are independent but are not necessarily identically distributed.

On the singularity probability of discrete random matrices

TL;DR

This work advances the understanding of singularity probabilities for discrete random matrices by proving a general exponential upper bound under a -boundedness framework and a structured value-set. The approach combines a finite-field reduction with a Structure Theorem akin to Freiman-type results, and employs Halász-type Fourier analysis to partition hyperplanes by combinatorial dimension into manageable cases. Consequently, it yields improved bounds, notably for Bernoulli entries, and extends to matrices with complex entries, non-identical distributions, and partially random structures, including results on rational eigenvalues for integer matrices. The methods provide a robust, broadly applicable toolkit for exponential bounds in discrete random matrix settings, with potential impact on random combinatorics and numerical linear algebra.

Abstract

Let be an by random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that is singular is at most , improving on the previous best upper bound of proven by Tao and Vu in arXiv:math/0501313v2. This paper follows a similar approach to the Tao and Vu result, including using a variant of their structure theorem. We also extend this type of exponential upper bound on the probability that a random matrix is singular to a large class of discrete random matrices taking values in the complex numbers, where the entries are independent but are not necessarily identically distributed.

Paper Structure

This paper contains 22 sections, 37 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The general theorem
  3. Some corollaries of Theorem \ref{['main theorem']}
  4. Proving Inequalities \ref{['result 1']}, \ref{['result 2']}, and \ref{['result 3']}
  5. Matrices with entries having symmetric distributions
  6. Random matrices with entries having arbitrary distributions
  7. Partially random matrices
  8. Integer matrices and rational eigenvalues
  9. Random matrices with complex entries: A reduction technique
  10. Proof of the main theorem (Theorem \ref{['main theorem']})
  11. Definitions and preliminaries
  12. Proof of the medium combinatorial dimension
  13. The unexceptional medium combinatorial dimension case
  14. Analyzing the exceptional medium combinatorial dimension case
  15. Halász-type arguments
  16. ...and 7 more sections

Key Result

Corollary 1.2

Let $p$ be a real constant between 0 and 1, let $c$ be any positive constant less than $1/\ln(1/p)$, and let $S\subset \mathbb C$ be a set of complex numbers having cardinality $\left|S\right|\le O(1)$. Let $N_{\mathfrak{f},n}$ be an $n$ by $n$ complex matrix in which $\mathfrak{f}\le c\ln n$ rows c

Figures (2)

  • Figure 1: Let $P(\mu):= \lim_{n\to\infty}\Pr \left(M_{\pm 1,n}^{(\mu)} \hbox{is singular}\right)^{1/n}$, where $M_{\pm 1,n}^{(\mu)}$ is the $n$ by $n$ matrix with independent random entries taking the value $0$ with probability $1-\mu$ and the values $+1$ and $-1$ each with probability $\mu/2$. The solid lines denote the upper bounds on $P(\mu)$ given by Inequalities \ref{['result 1']}, \ref{['result 2']}, and \ref{['result 3']}, and the dashed lines denote the lower bounds given by Inequalities \ref{['lower bound 1']} and \ref{['lower bound 2']}. The upper and lower bounds coincide for $0 \le \mu \le \frac{1}{2}$, and the shaded area shows the difference between the best known upper and lower bounds for $\frac{1}{2} \le \mu \le 1$. The straight line segments from the point $(0,1)$ to $(1/2,1/2)$ and from the point $(1/2,1/2)$ to $(1,3/4)$ represent the best upper bounds we have derived using the ideas in TV2, and the curve $1-2\mu+\frac{3}{2}\mu^2$ for $0\le\mu\le 1$ represents a sometimes-better upper bound we have derived by adding a new idea. Note that the upper bounds given here also apply to the singularity probability of a random matrix with independent entries having arbitrary symmetric distributions in a set $S$ of complex numbers, so long as each entry is 0 with probability $1-\mu$ and the cardinality of $S$ is $\left|S\right|\le O(1)$ (see Corollary \ref{['gen result']}).
  • Figure 2: Let $P(\mu):= \lim_{n\to\infty}\Pr \left(M_{\{\pm 2,\pm 1\},n}^{(\mu)} \hbox{is singular}\right)^{1/n}$, where $M_{\{\pm 2,\pm 1\},n}^{(\mu)}$ is the $n$ by $n$ matrix with independent random entries taking the value 0 with probability $1-\mu$ and the values $+2,-2,+1,-1$ each with probability $\mu/4$. This figure summarizes the upper bounds on $P(\mu)$ from Corollary \ref{['pm2case']} and the lower bounds from Displays \ref{['2lb1']} and \ref{['2lb2']}. The best upper bounds (shown in thick solid lines) match the best lower bounds (thick dashed lines) for $0 \le\mu\le \frac{16}{25}$; and it is not hard to improve the upper bound a small amount by finding a bound (of exponent 1) to bridge the discontinuity. One should note that even as stated above, the upper bounds are substantially better than those given by Corollary \ref{['gen result']} (which are shown in Figure \ref{['figure 1']}). The shaded area represents the gap between the upper and lower bounds.

Theorems & Definitions (68)

  • Conjecture 1.1
  • Corollary 1.2
  • Remark 1.3: Other exponential bounds
  • Theorem 1.4
  • Definition 2.1: $p$-bounded of exponent $r$
  • Theorem 2.2
  • Remark 2.3: Strict positivity in Inequality \ref{['pDqbounded']}
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • ...and 58 more