The Annals of Probability

Asymptotic density in a threshold coalescing and annihilating random walk

David Stephenson

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We consider an interacting random walk on $\mathbb{Z}^d$ where particles interact only at times when a particle jumps to a site at which there are at least $n - 1$ other particles present. If there are $i \ge n - 1$ particles present, then the particle coalesces (is removed from the system) with probability $c_i$ and annihilates (is removed along with another particle) with probability $a_i$. We call this process the $n$-threshold randomly coalescing and annihilating random walk. We show that, for $n \ge 3$, if both $a_i$ and $a_i + c_i$ are increasing in $i$ and if the dimension $d$ is at least $2n + 4$, then

$P\{\text{the origin is occupied at time $t$}\}\sim C(d, n) t^{1/(n-1)}$

$E\{\text{ number of particles at the origin at time $t$}\} C(d, n) t^{1/(n-1)}$

The constants $C(d, n)$ are explicitly identified. The proof is an extension of a result obtained by Kesten and van den Berg for the 2-threshold coalescing random walk and is based on an approximation for $dE(t)/dt$.

Article information

Ann. Probab., Volume 29, Number 1 (2001), 137-175.

First available in Project Euclid: 21 December 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60k35 82C22: Interacting particle systems [See also 60K35]

Coealescing random walk annihilating random walk particle density threshold


Stephenson, David. Asymptotic density in a threshold coalescing and annihilating random walk. Ann. Probab. 29 (2001), no. 1, 137--175. doi:10.1214/aop/1008956326.

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