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Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

Hoi H. Nguyen

Abstract

Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.

Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

Abstract

Let denote a random symmetric by matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that is non-singular with probability for any positive constant . The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.

Paper Structure

This paper contains 11 sections, 39 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The approach
  3. The inverse Littlewood-Offord problem for linear forms
  4. The inverse Littlewood-Offord problem for bilinear forms
  5. The inverse Littlewood-Offord problem for quadratic forms
  6. A rank reduction argument and the full rank assumption
  7. proof of Theorem \ref{['theorem:ILO:bilinear:differentGAP']} , Theorem \ref{['theorem:ILO:bilinear:commonGAP']}, and Theorem \ref{['theorem:ILO:bilinear:general']}
  8. proof of Theorem \ref{['theorem:ILO:quadratic:differentGAP']} and Theorem \ref{['theorem:ILO:quadratic:commonGAP']}
  9. proof of Lemma \ref{['lemma:normalvector:linear']}
  10. Proof of Lemma \ref{['lemma:normalvector:quadratic']}
  11. proof of Theorem \ref{['theorem:singularity:case1']} and Theorem \ref{['theorem:singularity:case2']}

Key Result

Theorem 1.2

We have for any positive constant $C$, where the implied constant depends on $C$.

Theorems & Definitions (70)

  • Conjecture 1.1
  • Theorem 1.2: Main theorem
  • Conjecture 1.3
  • Lemma 2.1
  • Lemma 2.2: Odlyzko's lemma,O
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • ...and 60 more