Random matrix theory over finite fields

Random Matrix Theory over Finite Fields

The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various meas...

Matrix Theory over Finite Fields : A Survey

Eigenvalues of Random Matrices over Finite Fields

We determine the limiting distribution of the number of eigenvalues of a random n×n matrix over Fq as n → ∞. We show that the q → ∞ limit of this distribution is Poisson with mean 1. The main tool is a theorem proved here on asymptotic independence for events defined by conjugacy class data arising from distinct irreducible polynomials. The proof of this theorem uses the cycle index for matrice...

On random polynomials over finite fields

We consider random monic polynomials of degree n over a finite field of q elements, chosen with all q possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is q. We establish various approximations for the joint distribu...

Random Krylov Spaces over Finite Fields

Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself. We obtain an exact formula for this probability, and from it we derive good lower bounds that approach 1 exponentiall...