[Paper Review] Random matrices over finite fields: methods and results

Kyle Luh, Sean Meehan+1 more|arXiv (Cornell University)|Jul 4, 2019

Random Matrices and Applications19 references4 citations

TL;DR

This paper develops advanced probabilistic methods to analyze the discrepancy of random walks modulo a prime p, applying these tools to study the rank distribution of random matrices over finite fields F_p and the equidistribution of normal vectors in random hyperplanes. It establishes universality results for eigenvalue-free matrices and characteristic polynomial divisibility, extending prior work by Stong and Neumann-Praeger beyond the uniform model.

ABSTRACT

In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.

Motivation & Objective

• To characterize random walks with large discrepancy modulo a prime p, especially when steps are non-identically distributed.
• To analyze the distribution of the rank of random matrices over F_p using discrepancy-based methods.
• To study the equidistribution of normal vectors of random hyperplanes in finite-dimensional vector spaces over F_p.
• To determine the limiting probability that a random square matrix over F_p is eigenvalue-free or has a characteristic polynomial divisible by a given irreducible polynomial.
• To extend universality results of Stong and Neumann-Praeger beyond the uniform model to broader classes of random matrix ensembles.

Proposed method

• Uses characterizations of discrepancy in random walks modulo p to bound deviations from uniformity.
• Applies Fourier-analytic and combinatorial techniques to model the distribution of matrix ranks and normal vectors.
• Leverages generating functions and moment methods to analyze the probability of eigenvalue-free matrices.
• Employs equidistribution theory over finite fields to study the likelihood of characteristic polynomial divisibility by irreducible polynomials.
• Introduces a framework that generalizes previous results by relaxing assumptions on the distribution of matrix entries.
• Uses asymptotic analysis as n → ∞ to derive limiting probabilities for key matrix properties in F_p.

Experimental results

Research questions

• RQ1 What conditions on non-uniform random walks modulo p lead to significant discrepancy from uniformity, and how can this be quantified?
• RQ2 How does the rank of a random matrix over F_p distribute when matrix entries are drawn from non-uniform, possibly dependent distributions?
• RQ3 To what extent are normal vectors of random hyperplanes equidistributed in F_p^n, and how does this depend on the underlying matrix distribution?
• RQ4 What is the limiting probability that a random n×n matrix over F_p has no eigenvalues in F_p as n → ∞?
• RQ5 How universal is the probability that the characteristic polynomial of a random matrix over F_p is divisible by a fixed irreducible polynomial, and does this extend beyond the uniform model?

Key findings

• The paper establishes that the rank distribution of random matrices over F_p is universal across a broad class of non-uniform entry distributions, extending previous results to non-uniform models.
• It proves that the normal vectors of random hyperplanes are equidistributed in F_p^n under mild moment conditions on the matrix entries, generalizing earlier uniformity results.
• The limiting probability that a random n×n matrix over F_p is eigenvalue-free tends to a universal constant as n → ∞, independent of the specific distribution of entries as long as they satisfy mild discrepancy bounds.
• The probability that the characteristic polynomial of a random matrix over F_p is divisible by a fixed irreducible polynomial converges to a universal limit as n → ∞, even when entries are non-uniform.
• The results demonstrate that key statistical properties of random matrices over F_p—such as rank, eigenvalue presence, and polynomial divisibility—are robust to non-uniformity in the entry distribution.
• The framework developed allows for the derivation of precise asymptotic probabilities for matrix properties, revealing deep universality in finite field matrix ensembles.