IS03: Self-Organized Criticality

Organizer: Balázs Ráth (Budapest University of Technology and Economics)

The particle density in mean-field Activated Random Walk and the 2D sandpile

Antal A. Jarai

We study the Activated Random Walk model with sleeping rate \(\lambda\) on the complete graph with \(N+1\) vertices. One the vertices serves as a ‘sink’, where particles are removed from the system. Starting with a configuration of only S particles, at discrete time steps an A particle is added, and the system is allowed to stabilise to a new all-S state. We show that at stationarity, the number of particles in the system is \[\rho_c(\lambda) N + a(\lambda) \sqrt{N \log N} (1+o(1)), \quad \text{in probability, as $N \to \infty$,}\] where the critical density \(\rho_c(\lambda) = \lambda/(1+\lambda)\), and \(a(\lambda) = \sqrt{\lambda}/(1+\lambda)\).
(Joint work with C. Mönch and L. Taggi.)

 
We contrast the above with results on the 2D Abelian sandpile, where the exact form of the lower order correction to the critical density is still open.
(Joint work with M.W. Elvidge.)