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 2θ _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.