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A single permutation, seen as union of disjoint cycles, represents a regulargraph of degree two. Consider $d$ many independent random permutations andsuperimpose their graph structures. It is a common model of a random regular(multi-) graph of degree $2d$. We consider the following dynamics. Thedimension (i.e. size) of each permutation grows by coupled Chinese RestaurantProcesses, while in time each permutation evolves according to the randomtransposition chain. Asymptotically in the size of the graph one observes aremarkable evolution of short cycles and linear eigenvalue statistics indimension and time. In dimension, it was shown by Johnson and Pal (2014) thatcycle counts are described by a Poisson field of Yule processes. Here, we givea Poisson random surface description in dimension and time of the limitingcycle counts for every $d$. As $d$ grows to infinity, the fluctuation of thelimiting cycle counts, across dimension, converges to the Gaussian Free Field.In time this field is preserved by a stationary Gaussian dynamics. The laws ofthese processes are similar to eigenvalue fluctuations of the minor process ofa real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationarystochastic processes.

The Random Transposition Dynamics on Random Regular Graphs and the Gaussian Free Field