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.)

The density conjecture for activated random walk

Tobias Johnson

Physicists Bak, Tang, and Wiesenfeld in the 1980s proposed “self-organized criticality” as an explanation for why systems in nature with no obvious phase transition can exhibit self-similarity and power-law tails reminiscent of statistical mechanics systems at criticality. Based on simulations, they and others proposed that simple mathematical models of sandpiles drive themselves to criticality, in various senses. These predictions have been quite difficult to confirm mathematically. We consider activated random walk (ARW), a sandpile model that seems to have some universality. In dimension one, we prove the density conjecture: ARW on a finite interval with particles added in the middle and destroyed at the edges naturally drives itself to the critical density of ARW on the infinite line. This is the first rigorous proof of any sandpile model driving itself to a critical state. We also prove that the system mixes as soon as enough particles are added to bring the density up to criticality. Joint work with Chris Hoffman, Matt Junge, and Josh Meisel.

Self-organized criticality and avalanches in 2D forest fires

Pierre Nolin

Bernoulli percolation can be used to analyze planar forest fire (or epidemics) processes. In such processes, all vertices of a lattice are initially vacant, and then become occupied at rate \(1\). If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate \(\zeta\), all the vertices connected to it burn immediately, i.e. they become vacant.   We want to analyze the behavior of such processes near and beyond criticality, that is, when large components of occupied sites appear. They display a form of self-organized criticality as \(\zeta \searrow 0\), where the phase transition of Bernoulli percolation plays an important role. In particular, a peculiar and striking phenomenon arises, that we call “near-critical avalanches”.   This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and with Wai-Kit Lam (National Taiwan University, Taipei).