A Probabilistic Approach to Conjugacy Classes in the Finite Symplectic and Orthogonal Groups

Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods, one obtains a probabilistic proof of Steinbergʹs count of unipotent matrices and generalizations of formulas of Rudvalis and Shinoda.