# Multi-particle random walks

Define a multi-particle "breeding" random walk $\mathcal{W_p}$ in $d$ dimensions, for $p \in (0,1)$ as follows:

• The state of $\mathcal{W_p}$ at integer time $t\geq 0$ consists of the pair $(k, x)$ where $k \in \Bbb{Z}^+ \cup 0$ and $x \in \left(\Bbb{Z}^d \right)^k$. Informally, the state at time $t$ consists of zero or more particles, each on a node of a $d$-dimensional rectilinear grid.

• The state of $\mathcal{W_p}$ at $t=0$ is $(1, 0^d)$, that is, a single particle at the origin.

• The transition rules from the state at time $t$ to that at time $t+1$ are described in two phases: the first phase is movement and breeding, in which for each particle (that is, for each of the $k$ projections of element of $x$, which in turn are elements of $\Bbb{Z}^d$), with probability $1-p$ that particle moves with equal probability to each of the $2\cdot d$ neighboring vertices in the grid (as in a usual unbiased random walk), and with probability $p$ splits into two particles, which move to the two neighbors along a uniformly randomly selected direction among the $d$ dimensions.

• The second phase of the transition consists of dealing with cases in which multiple particles end up at the same node. Three separate models are of interest:

(a) The annihilating multiparticle walk, where the transition rule states that if two or more particles arrive at the same node at time $t$, all of them are removed from the state for time $t+1$. An annihilating multiparticle walk may reach a stopping position with zero particles, which we call "extinction."

(b) The limited multiparticle walk, where the transition rule states that if two or more particles arrive at the same node at time $t$, all but one of them are removed from the state for time $t+1$. The state in this model could equally well be described as a finite subset of $\Bbb{Z}^d$, rather than specifying the number of particles and listing the nodes of each particle.

(c) The exploding multiparticle walk, in which the second phase of the transition is trivial, and multiple particles can occupy the same node.

Have these models been studied? Of particular interest are two questions:

For a one-dimensional annihilating multiparticle walk, what is the probability of eventual extinction as a function of $p$?

Define a "return to the origin" as a time $t$ at which there is at least one particle at the origin (it is allowed that other particles may be elsewhere at that time). For a three-dimensional exploding random walk, for what (if any) values of $p$ does $\mathcal{W_p}$ almost surely return to the origin?