Random matrix central limit theorems for nonintersecting random walks
  We consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function. We prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity, several limiting distributions of the walks at the mid-time behave as the eigenvalues of random Hermitian matrices as the dimension of the matrices grows to infinity. 
Jinho Baik, Toufic M. Suidan