Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of these matrices as $n \tends \infty$, when the test functions of the statistics are polynomials. The proofs are mainly combinatorial, based on the trace formula, method of moments and some results on process convergence.