Random Matrices and Brownian Motion

Source abstract

For T ∈ GL n ( F q ), let Ω n ( t, T ) be the number of irreducible factors of degree less than or equal to n t in the characteristic polynomial of T . Let and suppose T is chosen from G L n ( F q ) at random uniformly. We prove that the stochastic process ≺ Z n ( t )≻ t ∈[0, 1] converges to the standard Brownian motion process W ( t ), as n → ∞.