Quadrature Domain Limit Shape for Activated Random Walk in One Dimension

Abstract

We study one-dimensional activated random walk (ARW) with sleep rate $\lambda>0$, started from one active particle at each site of a bounded open set $A\subset\mathbb{R}$. For such initial configurations we prove that, after rescaling, the sleeping particles concentrate on an open set $A^*\subset\mathbb{R}$ determined by a quadrature inequality for concave test functions, with asymptotic particle density $\rho_c(\lambda)$ on $A^*$. This confirms the quadrature inequality Conjecture~6 of Levine and Silvestri for one-dimensional ARW.