The random transposition dynamics on random regular graphs and the Gaussian free field

A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider d many independent random permutations and superimpose their graph structures. This is the well known permutation model of a random regular (multi-) graph of degree 2d. We consider a two dimensional field of d permutations indexed by size and time. The size of each permutation grows by coupled Chinese Restaurant Processes, while in time, each permutation evolves according to the random transposition chain. Via the permutation model, this projects to give a two parameter family of graphs growing in size ("dimension") and evolving over time. Asymptotically in this random graph ensemble one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. In dimension, it was shown by Johnson and Pal (Ann. Probab. 42 (2014) 1396-1437) that cycle counts are described by a Poisson field of Yule processes. Here, we give a Poisson random surface description in dimension and time of the cycle process, for every d. As d grows to infinity, the fluctuation of the limiting cycle counts, converges to a Gaussian process indexed by dimension and time. The marginal along dimension turns out to be the Gaussian Free Field and the process is stationary in time. Similar covariance structure appears in eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes. Thus this article describes a Poisson analogue of a natural Markovian dynamics on the Gaussian free field and its path properties.