Activated random walk on the complete graph has stationary particle count concentrating at ρ_c N + a √(N log N) with ρ_c = λ/(1+λ) and a = √λ/(1+λ).
and Sidoravicius, Vladas , TITLE =
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The stabilized single-source stochastic sandpile with p-topplings has macroscopic limit shape equal to a symmetric interval around the origin, with rescaled boundary fluctuations converging to a Gaussian.
On infinite bounded-degree graphs, divisible sandpiles with i.i.d. initial masses of mean μ stabilize almost surely if μ < 1 and masses have finite p-moment for p > 3, but explode if μ ≥ 1; the conditions are nearly sharp via counterexamples on other graphs.
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The number of particles in activated random walk on the complete graph
Activated random walk on the complete graph has stationary particle count concentrating at ρ_c N + a √(N log N) with ρ_c = λ/(1+λ) and a = √λ/(1+λ).
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Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$
The stabilized single-source stochastic sandpile with p-topplings has macroscopic limit shape equal to a symmetric interval around the origin, with rescaled boundary fluctuations converging to a Gaussian.
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Divisible sandpiles via random walks in random scenery
On infinite bounded-degree graphs, divisible sandpiles with i.i.d. initial masses of mean μ stabilize almost surely if μ < 1 and masses have finite p-moment for p > 3, but explode if μ ≥ 1; the conditions are nearly sharp via counterexamples on other graphs.