Random matrix theory over finite fields

Abstract

Abstract. 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 measures on partitions.

Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

“…Similarly, I lament not describing the many interactions with algebraic combinatorics. See [4], [43], [51]. Many of the results stated here for unitary matrices are "universal", applying to many other matrix ensembles (just as the central limit theorem): see Tracy-Widom [93] for an overview of the many great results.…”

“…It is an empirical fact that the general question seems to lead to elegant mathematics which has surprisingly useful consequences. A marvelous survey, making just these points and verifying them on finite groups of Lie type, is given in Fulman [43].…”

Birational permutations

“…(2) does not have any indeterminacy point in Q(K); (3) is the identity at the north pole N (fixing the two lines contained in Q(K ) through N pointwise).…”

GL(n, q) and Increasing Subsequences in Non-Uniform Random Permutations

“…Thus one has a very natural probability measure on the set of all partitions of all natural numbers. Further discussion of this measure can be found in the survey [9]. For our purposes we need the formula which says that the chance that this limit measure (which we denote P q ) yields λ is…”

“…It is possible to exactly sample from all four measures P q , Q q , P n, q , Q n, q . See [9] for discussion. The cases P n, q , Q n, q are joint work with Mark Huber.…”