We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the 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, arising from the study of multiple random walks on a random regular graph G. For k independent walks on 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 θ r n/ 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 ln k k θ r n, 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 (2 ln 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 same as the expected time for all the particles to coalesce, which is asymptotic to 2θ r n.
Later, the authors of REF studied meet-exchange for the case of random regular graphs and k ⩽ n ε random walks.
Multiple Random Walks in Random Regular Graphs
in SIAM J. Discrete Math
Mathematics Computer Science