We study the long-time behavior of conservative interacting particle systems in Z: the activated random walk model for reaction-diffusion systems and the stochastic sandpile. We prove that both systems undergo an absorbing-state phase transition.This preprint has the same numbering for sections, theorems, equations and figures as the published article "Invent. Math. 188 (2012): 127-150."

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

The Box-Ball System was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Meanwhile, solitons of different sizes interact through a momentary change of speeds during collision, which cumulatively affects their asymptotic speeds, suggesting that the speeds are determined by such interaction. In order to understand general invariant measures and soliton interactions, we introduce a decomposition of configurations through slots, reducing the dynamics to a simple hierarchical translation of different components. Using this property we obtain an explicit recipe to construct a rich family of invariant measures. Finally, we obtain explicit equations for the soliton speeds in terms of spacial density of solitons. BBS dynamics for i.i.d. initial configuration with density 0.25. Straight red lines are deterministic and computed using Theorem 1.2.

In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including Z d , and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate λ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case λ < +∞ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.