# Multiple Random Walks and Interacting Particle Systems

## Abstract

We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make our analysis for random regular graphs. The cover time of a random walk on a random *r*-regular graph was studied in [6], where it was shown with high probability (**whp**), that for *r* ≥ 3 the cover time is asymptotic to *θ*_{r}*n* ln *n*, where *θ*_{r} = (*r* − 1)/(*r* − 2). In this paper we prove the following (**whp**) results. For *k* independent walks on a random regular graph *G*, the cover time *C*_{G}(*k*) is asymptotic to *C*_{G}/*k*, where *C*_{G} is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is \(\theta_r n/\binom{k}{2}\). If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for *k* walks to broadcast a single piece of information to each other is asymptotic to 2*θ*_{r}*n* (ln *k*)/*k*, as *k*,*n* → ∞.

We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing, or by annihilating each other. For example, the expected extinction time of *k* explosive particles (*k* even) tends to (2ln 2) *θ*_{r}*n* as *k*→ ∞.

The case of *n* coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The expected time for a unique opinion to emerge is the expected time for all the particles to coalesce, which is asymptotic to 2 *θ*_{r}*n*.

Combining results from the predator-prey and multiple random walk models allows us to compare expected detection time in the following cops and robbers scenarios: both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, the predators move randomly and the prey stays fixed. In all cases, with *k* predators and ℓ prey the expected detection time is *θ*_{r}*H*_{ℓ}*n*/*k*, where *H*_{ℓ} is the ℓ-th harmonic number.

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