17 problems

Let $\rho_{\mathtt{FE}}$ be the critical density of the fixed-energy activated random walk on $\mathbb{Z}^d$, depending on the dimension $d$ and sleep rate $\lambda$, and let…

Levine–Liang mixing-time conjecture. The mixing time is

Let driven-dissipative activated random walk (ARW) have a limiting density, and let the fixed-energy model have a critical value, as characterized by the corresponding fixed-energy…

For each $n$, let $4\mathbb P_n4$ be the law of the final sleeping configuration in $4\{0,\mathfrak{s}\}^{\mathbb Z^d}4$ for activated random walk on $4\mathbb Z^d4$ started with…

Sharp-bound conjecture. The upper bound

Consider activated random walk with an initial number of particles per site given by a random variable, and let $m$ denote the odometer. The expected odometer satisfies a lower bou…

Let $m(0)$ denote the odometer at the origin for fixed-energy activated random walk with i.i.d. initial particle density $\rho$, and let $\rho_c(\lambda)$ be its critical density f…

Let $p_\mathfrak{s}:=\lambda/(1+\lambda)$ be the sleep probability, let $\rho_\mathrm{c}$ denote the critical density for activated random walk on $\mathbb{Z}^d$, and let $d$ be th…

Ordering conjecture. These limits exist and satisfy

Levine–Liang's cutoff conjecture. With uniform driving, the process mixes as quickly as possible: mixing occurs as soon as the system density reaches $\rho_{\mathtt{DD}}$.

Let $u$ be the uniform driving sequence on $B_r = B(0,r) \cap \mathbb{Z}^d$. Let $0 < \lambda < \infty$ be a constant sleep rate, and let $P$ be the simple random walk on $B_r$ wit…

Outer-critical density conjecture.

Density equality conjecture.

In one-dimensional Activated Random Walk with sleep rate $\lambda>0$, let $\zeta_a(\mathbb Z,\lambda)$ denote the aggregate density from the single-source experiment, and let…

Let ${\rm ARW}(\mu,\lambda)$ be activated random walk on $\mathbb{Z}$, with sleep rate $\lambda>0$, and let $\mu_c(\lambda)$ denote its critical mass density. Vanishing critical-de…

Let ${\rm ARW}(\mu,\lambda)$ be activated random walk on $\mathbb{Z}$, with sleep rate $\lambda>0$, and let $\mu_c(\lambda)$ denote its critical mass density. Critical-density boun…

Consider activated random walk on the one-dimensional lattice $\mathbb{Z}$ with sleeping rate $\lambda$, jump distribution $p(1)=q$ and $p(-1)=1-q$, where $q\in[0,1]$, and an initi…