DIP-RAMP-PLATEAU FOR DYSON BROWNIAN MOTION FROM THE IDENTITY ON U(N)

Abstract

In recent work, the authors have shown that the eigenvalue probability density function for Dyson Brownian motion from the identity on U(N) is an example of a newly identified class of random unitary matrices called cyclic Pólya ensembles. In general, the latter exhibit a structured form of the correlation kernel. Specialising to the case of Dyson Brownian motion from the identity on U(N) allows the moments of the spectral density, and the spectral form factor SN (k; t ), to be evaluated explicitly in terms of a certain hypergeometric polynomial. Upon transformation, this can be identified in terms of a Jacobi polynomial with parameters (N(µ − 1), 1), where µ = k/N and k is the integer label..