Critical density of activated random walks on transitive graphs

We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density .c for sustained activity is strictly between 0 and 1. It was known that .c>0 on Zd, d.1, and that .c<1 on Z for small enough sleeping rate. We show that .c.0 as ..0 in all vertex-transitive transient graphs, implying that .c<1 for small enough sleeping rate. We also show that .c<1 for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that .c>0 in any vertex-transitive amenable graph, and that .c.(0,1) for any sleeping rate on regular trees.