Activated Random Walkers: Facts, Conjectures and Challenges

Abstract

We study a particle system with hopping (random walk) dynamics on the integer lattice Z d . The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ > 0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β = 1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also

Conserved directed percolation: exact quasistationary distribution of small systems and Monte Carlo simulations

“…Here we consider initial configurations in which N particles (all active) are distributed randomly amongst the sites, respecting the prohibition of multiple occupancy. (In the ARW model [14] the number of particles per site is unrestricted but only an isolated particle can sleep. )…”

“…We study the model in extensive Monte Carlo simulations as well, in efforts to better characterize CDP critical behavior. A closely related model, activated random walkers (ARW), was introduced in [14]; in this case there is no restriction on the number of walkers per site. In [14] the principal emphasis was on asymmetric ARW (hopping in one direction only); some preliminary evidence for CDP-like behavior of the symmetric version was also reported.…”

Asymptotic Shape and Propagation of Fronts for Growth Models in Dynamic Random Environment

“…It remains a challenging problem to prove the analog of the Theorem 9 in dimensions two and more. For general account on this process and other interesting open problems we refer to [9].…”

Activated random walk on a cycle

“…Each sleepy particle is awakened when an active particle occupies the same site. This model, known as Activated Random Walk (ARW), has attracted interest in non-equilibrium statistical mechanics as well as probability literature in recent years in connection with studying fixed energy sandpile models [7,28,29,5,30,8,22]. The motivation of studying this model is two-fold.…”