arXiv subjects

Harley Kaufman

Publications and source records attributed to Harley Kaufman.

3 recordsLinked to original sources

arXiv2026

Scaling limit and density conjecture for activated random walk on the complete graph

We study driven-dissipative activated random walk with sleep probability $p$ on an $n$-vertex complete graph with a sink that traps jumping particles with probability $q_n$. We show that the number of sleeping particles $S_n$ left by the stationary distribution has a Gumbel scaling limit for $\exp(-n^{1/3}) \ll q_n \ll n^{-1/2}$. The particular scaling implies that $S_n$ is hyperuniform and thus the stationary configuration law has negative correlations and is not a product measure. We also prove that $S_n/n$ converges to $p$ if and only if $q_n = e^{-o(n)}$, and that, when $q_n=0$, the number of jumps to stabilization undergoes a phase transition at density $p$.

math.PR↗

arXiv2025

Asymptotic behavior of the critical density of activated random walk

We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $\lambda$ tends to $0$ and $\infty$. For large $\lambda$, we prove new lower bounds in dimensions 1 and 2, showing that in one dimension the critical density approaches $1$ superpolynomially fast. For small $\lambda$, we prove a new lower bound in two dimensions for how fast the critical density vanishes. We also obtain the first-order approximation for transient walks in both regimes.

math.PR↗

arXiv2025

Mean-field limit for activated random walk on the integer lattice

We show that the critical density for activated random walk on $\mathbb{Z}^d$ approaches the sleep probability as $d \to \infty$ and provide the first-order correction.

math.PR↗