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Invertibility for non-Hermitian and symmetric random band matrices with sublinear bandwidth and discrete entries

Yi Han

Abstract

A well-known result in random matrix theory, proven by Kahn, Komlós and Szemerédi in 1995, states that a square random matrix with i.i.d. uniform $\{\pm 1\}$ entries is invertible with probability $1-\exp(-Ω(n))$. As a natural generalization of the model, we consider the invertibility of a class of random band matrices with independent entries where the bandwidth $d_n$ scales like $n^α$, for some $α\in(0,1)$. The band matrix model we consider is sufficiently general and covers existing models such as the block band matrix and periodic band matrix, allowing great flexibility in the variance profile. As the bandwidth is sublinear in the dimension, estimating the invertibility and least singular values of these matrices is a well-known open problem. We make progress towards the invertibility problem by showing that, when $α>\frac{2}{3}$ and when the random variables are i.i.d. uniformly distributed on $\{\pm 1+c\}$ for any fixed integer $c$, then the band matrix is invertible with probability $1-\exp(-Ω(n^{α/2}))$. Previously, even invertibility with probability $1-o(1)$ was not known for these band matrix models except in the very special case of block band matrices. We then extend the invertibility result to symmetric random band matrices with integer entries, and prove the same non-singularity probability estimate whenever $α>\frac{2}{3}$.

Invertibility for non-Hermitian and symmetric random band matrices with sublinear bandwidth and discrete entries

Abstract

A well-known result in random matrix theory, proven by Kahn, Komlós and Szemerédi in 1995, states that a square random matrix with i.i.d. uniform entries is invertible with probability . As a natural generalization of the model, we consider the invertibility of a class of random band matrices with independent entries where the bandwidth scales like , for some . The band matrix model we consider is sufficiently general and covers existing models such as the block band matrix and periodic band matrix, allowing great flexibility in the variance profile. As the bandwidth is sublinear in the dimension, estimating the invertibility and least singular values of these matrices is a well-known open problem. We make progress towards the invertibility problem by showing that, when and when the random variables are i.i.d. uniformly distributed on for any fixed integer , then the band matrix is invertible with probability . Previously, even invertibility with probability was not known for these band matrix models except in the very special case of block band matrices. We then extend the invertibility result to symmetric random band matrices with integer entries, and prove the same non-singularity probability estimate whenever .

Paper Structure

This paper contains 12 sections, 18 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Invertibility for random symmetric band matrices
  3. The proof for the independent case
  4. Finite field inverse Littlewood-Offord counting problem
  5. The main argument for Theorem \ref{['maintheorem']}
  6. Invertibility for the symmetric band matrix model
  7. Justifying shapes of null vectors of submatrices
  8. Eliminating kernel vectors of structure type I
  9. Eliminating kernel vectors of structure type II
  10. Eliminating kernel vectors of structure type III
  11. Eliminating kernel vectors of structure type IV
  12. Linear algebraic reductions and invertibility

Key Result

Theorem 1.1

Fix $\alpha\in(0,1)$ and let $d_n$ be a sequence of integers such that $d_n\geq n^\alpha$ when $n$ is sufficiently large. Also assume that $d_n\leq n^{1-\beta}$ for some small $\beta>0$. Let $A_n=(a_{ij})_{1\leq i,j\leq n}$ be a family of square random matrices with independent entries. We assume th Then whenever $\alpha>\frac{2}{3}$, we have for some universal constant $C>0$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 38 more