13 problems

Start activated random walks on $\mathbb{Z}^d$ with $k$ active particles at the origin and no particles elsewhere, and let $A_k$ be the aggregate, namely the set of sites visited a…

Let $V_n={1,\dots,n}subset\mathbb{Z}$, and consider the driven-dissipative Markov chain on stable configurations on $V_n$ obtained by repeatedly adding an active particle at a unif…

Let $d\geq 1$, start with $n$ particles at the origin of $\mathbb{Z}^d$, and stabilize them in $\mathbb{Z}^d$. Denote by $A_n$ the random set of visited sites, and let $\zeta_c$ be…

Let $\mu_c(\lambda)$ denote the critical particle density for the Activated Random Walk model on $\mathbb{Z}^d$ with sleep rate $\lambda$. Active-phase conjecture. For every finite…

Consider the Activated Random Walk model on a finite box of a vertex-transitive locally finite graph, in which particles are added one at a time at uniformly chosen sites, particle…

Density conjecture. The critical density and the self-organized critical density should coincide:

Consider the activated random walk model on $\mathbb{Z}^d$ for $d\geq 3$, with particle density and sleeping rate as its parameters. Strictly subunit critical density conjecture. T…

Let a system evolve according to the Activated Random Walk model on a general class of graphs, including $\mathbb Z^d$, from general initial conditions. The critical point is the t…

Let $d\geqslant 1$, let the particles perform any translation-invariant random walk, and let $\lambda>0$. For i.i.d. Poisson initial data of density $\zeta$, let $\zeta_c(\lambda)$…

Consider activated random walks with a family of initial particle-number distributions $\{P_\mu\}_{0\leqslant\mu<\infty}$ on $\{0,1,2,\dots\}$, parametrized by their first moment…

Consider activated random walks on $\mathbb{Z}$ with finite-range random-walk jumps, parameter $\lambda>0$, and critical density $\mu_c$. The source establishes only that $\mu_c>0$…

Consider the activated random walk model on $\mathbb{Z}^d$ with fixed parameter $\lambda>0$, and let $\mu_c$ be its critical density, so that the system fixates for $\mu<\mu_c$ and…

Consider the activated random walk model on $\mathbb{Z}^d$ with activity parameter $\lambda>0$ and i.i.d. initial numbers of active particles having expectation $\mu$. The system l…