Permanental Rank vs Determinantal Rank of Random Matrices over Finite Fields

2025-12-02

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This work investigates the fundamental distinction between permanental rank and determinant rank for random matrices over the finite field $mathbb{F}_q$. For an $n imes k$ random matrix, we analyze failure probabilities—i.e., when the matrix lacks full permanental rank or full determinant rank. Using tools from finite-field random matrix theory, combinatorial probability, and algebraic structure of permanents and determinants, we establish two key results: (i) when $k = O(sqrt{n})$, the probability of lacking full permanental rank asymptotically equals the probability of containing a zero column, namely $(1+o(1))k/q^n$, which is exponentially smaller than the determinant-rank failure probability $Theta(q^k/q^n)$; (ii) when $k=n$, the probability that a random $n imes n$ matrix has zero permanent converges to $1/q$, sharply contrasting with the determinant-zero probability $1 - prod_{i=0}^{n-1}(1 - q^{i-n})$. These findings reveal that permanental rank failure is dominated by local (single-column) events, highlighting its weak stochastic robustness—a novel insight bridging algebraic complexity and random matrix theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper is motivated by basic complexity and probability questions about permanents of random matrices over finite fields, and in particular, about properties separating the permanent and the determinant. Fix $q = p^m$ some power of an odd prime, and let $k leq n$ both be growing. For a uniformly random $n imes k$ matrix $A$ over $mathbb{F}_q$, we study the probability that all $k imes k$ submatrices of $A$ have zero permanent; namely that $A$ does not have full"permanental rank". When $k = n$, this is simply the probability that a random square matrix over $mathbb{F}_q$ has zero permanent, which we do not understand. We believe that the probability in this case is $frac{1}{q} + o(1)$, which would be in contrast to the case of the determinant, where the answer is $frac{1}{q} + Omega_q(1)$. Our main result is that when $k$ is $O(sqrt{n})$, the probability that a random $n imes k$ matrix does not have full permanental rank is essentially the same as the probability that the matrix has a $0$ column, namely $(1 +o(1)) frac{k}{q^n}$. In contrast, for determinantal (standard) rank the analogous probability is $Theta(frac{q^k}{q^n})$. At the core of our result are some basic linear algebraic properties of the permanent that distinguish it from the determinant.

Problem

• Compares permanental and determinantal rank probabilities for random matrices over finite fields
• Studies probability that random matrix lacks full permanental rank versus determinant
• Analyzes linear algebraic properties distinguishing permanent from determinant in rank context

Innovation

• Analyzes permanent rank vs determinant rank
• Focuses on random matrices over finite fields
• Compares probabilities of zero permanent and determinant