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

TL;DR

This paper studies the evolution of cycle counts in random regular graphs generated by multiple permutations, revealing a Poisson surface description and convergence to the Gaussian Free Field as the degree increases.

Contribution

It introduces a Poisson random surface model for cycle counts in random regular graphs and shows their convergence to the Gaussian Free Field as the degree grows.

Findings

Cycle counts are described by a Poisson random surface in dimension and time.

As degree increases, fluctuations converge to the Gaussian Free Field.

The evolution in time is governed by a stationary Gaussian dynamics.

Abstract

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. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The dimension (i.e. size) of each permutation grows by coupled Chinese Restaurant Processes, while in time each permutation evolves according to the random transposition chain. Asymptotically in the size of the graph 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 (2014) 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 limiting cycle counts for every $d$. As $d$ grows to infinity, the fluctuation of the limiting cycle counts, across dimension, converges to the Gaussian Free Field. In time this field is preserved by a stationary Gaussian dynamics. The laws of these processes are similar to eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes.