The critical window in activated random walk on the complete graph

Abstract: We consider an elementary model for self-organised criticality, the activated random walk on the complete graph. We introduce a discrete time Markov chain as follows. At each time step we add an active particle at a random vertex and let the system stabilise following the activated random walk dynamics, obtaining a particle configuration with all sleeping particles. Particles visiting a boundary vertex are removed from the system. We characterise the support of the stationary distribution of this Markov chain, showing that, with high probability, the number of particles concentrates around the value $\rho_c N + a \sqrt{N \log N}$ within a critical window of size at most $o( \sqrt{N \log N })$, where $N$ is the number of vertices. Due to the mean-field nature of the model, we are able to determine precisely the critical density $\rho_c= \frac{\lambda}{1+\lambda}$, where $\lambda$ is the sleeping rate, as well as the constant $a = \frac{\sqrt{\lambda}}{1+\lambda}$ characterising the lower order shift. Our approach utilises results about super-martingales associated with activated random walk that are of independent interest.