Macroscopic flow out of a segment for Activated Random Walks in dimension 1

TL;DR

This work analyzes Activated Random Walks in one dimension, establishing that in the supercritical regime a segment stabilized with exits forbidden outside the segment will have a positive fraction of particles exit with positive probability, and that this exit property is essentially necessary for sustained activity. The authors develop a site-wise (Abelian) representation, construct a block configuration and a coarse-grained coupling to a directed, origin-focused ARW, and leverage RS12 results to link finite-box behavior to infinite-lattice phase transitions. They prove a precise equivalence: $\zeta>\zeta_c$ if and only if the stabilized segment reveals a positive exit fraction, and provide explicit probabilistic bounds on the no-exit event, along with an explicit quantitative bound relating exit probabilities to $\lambda$ and $\zeta$. The results connect finite-volume stabilization to the global active phase, address conjectures such as hockey-stick and ball, and clarify the critical behavior in dimension 1, with implications for self-organized criticality and universality of $\zeta_c$ in this setting.

Abstract

Activated Random Walk is a system of interacting particles which presents a phase transition and a conjectured phenomenon of self-organized criticality. In this note, we prove that, in dimension 1, in the supercritical case, when a segment is stabilized with particles being killed when they jump out of the segment, a positive fraction of the particles leaves the segment with positive probability. This was already known to be a sufficient condition for being in the active phase of the model, and the result of this paper is that this condition is also necessary, except maybe precisely at the critical point. This result can also be seen as a partial answer to some of the many conjectures which connect the different points of view on the phase transition of the model.

Figures (4)

• Figure 1: Outline of the proof of Theorem \ref{['thm_no_exit']}: assuming that a given deterministic initial configuration $\eta:V_n\to\mathbb{N}$ produces a good stabilization with probability $p$, we build a block configuration on $\mathbb{Z}$ with density $q\zeta$ which fixates if $q<p$, showing that $q\zeta\leq \zeta_c$. Since this holds for every $q<p$, we get $p\zeta\leq\zeta_c$.
• Figure 2: Coupling between the stabilization of the block configuration, on the left side, and the stabilization of a coarse-grained configuration, on the right side (we refer to Section \ref{['subsec_sketch_no_exit']} for an informal sketch and to Proposition \ref{['prop_coupling']} for a more detailed presentation of the coupling).
• Figure 3: The two steps of the proof of Theorem \ref{['thm_explicit']}.
• Figure 4: Strategy to prove Lemma \ref{['lemma_NML']} in the case $\max W=\max V_n$. The key point is that we always topple the leftmost active particle in $V_n$, so that whenever a particle jumps out of $V_n$ from the left, all the other particles are active, allowing us to force this particle to walk out of $W$ with no effect on the other particles.

Theorems & Definitions (20)

• Theorem 1: RSZ19
• Theorem 2: RT18
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 1
• Theorem 6: Theorem 2 in RS12
• Lemma 1: Abelian property, Lemma 2 in RS12
• Lemma 2: Lemma 2.1 in Rolla20
• Proposition 2