2026-07-21T10:06:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:24403 2025-09-15T05:41:12Z ddc:004 open_access

Permanental Rank vs Determinantal Rank of Random Matrices over Finite Fields Ghasemi, Fatemeh Gross, Gal Kopparty, Swastik Permanent random matrices over a finite field This paper is motivated by basic complexity and probability questions about permanents of random matrices over small finite fields, and in particular, about properties separating the permanent and the determinant. Let q be a fixed odd prime, and let k ≤ n both be growing. For a uniformly random n × k matrix A over 𝔽_q, we study the probability that all k × 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 𝔽_q has zero permanent, which we do not understand. We believe that the probability in this case is 1/q + o(1), which would be in contrast to the case of the determinant, where the answer is 1/q + Ω_q(1). Our main result is that when k is O(√n), the probability that a random n × 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)) k/qⁿ. In contrast, for determinantal (standard) rank the analogous probability is Θ(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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Fatemeh Ghasemi and Gal Gross and Swastik Kopparty 2025 Is Part Of LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2025.37 urn:nbn:de:0030-drops-244037 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.37 eng https://creativecommons.org/licenses/by/4.0/legalcode