  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 stationary
distribution of this Markov chain, showing that, with high probability, the
number of particles concentrates around the value $\rho_c N$ 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 also able to
determine precisely the critical density $\rho_c=\lambda/(1+\lambda)$, where
$\lambda$ is the sleeping rate.
