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Permanents of random matrices over finite fields

Zach Hunter, Matthew Kwan, Lisa Sauermann

Abstract

Fix a finite field $\mathbb F_q$ and let $A\in \mathbb F_q^{n\times n}$ be a uniformly random $n\times n$ matrix over $\mathbb F_q$. The asymptotic distribution of the determinant $\det(A)$ is well-understood, but the asymptotic distribution of the permanent $\operatorname{per}(A)$ is still something of a mystery. In this paper we make a first step in this direction, proving that $\operatorname{per}(A)$ is significantly more uniform than $\det(A)$.

Permanents of random matrices over finite fields

Abstract

Fix a finite field and let be a uniformly random matrix over . The asymptotic distribution of the determinant is well-understood, but the asymptotic distribution of the permanent is still something of a mystery. In this paper we make a first step in this direction, proving that is significantly more uniform than .
Paper Structure (7 sections, 14 theorems, 58 equations)

This paper contains 7 sections, 14 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. General distributions
  3. Overview of the paper and proofs
  4. Proof for the uniform case
  5. Proof for general distributions
  6. Growing many matrices with nonzero permanents
  7. Approximate uniformity for linear maps

Key Result

Theorem 1.1

Fix a finite field $\mathbb{F}_{q}$ of odd characteristic. For a uniformly random $n\times n$ matrix $A\in\mathbb{F}_{q}^{n\times n}$, we have for all $n$, and we have the strict inequality Moreover, for all $n\ge 3$ we have for some absolute constant $C$.

Theorems & Definitions (34)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Remark
  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • ...and 24 more