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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  →  ∞    in    F  p   . We show that these statistics are universal, extending results of Stong and Neumann‐Praeger beyond the uniform model.

Some new results in random matrices over finite fields