Multiple Random Walks and Interacting Particle Systems

We study properties of multiple random walks on a graph under variousassumptions 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 studiedin [6], where it was shown with high probability (whp), that for r ≥ 3 thecover time is asymptotic to θrn lnn, where θr = (r − 1)/(r − 2).In this paper we prove the following (whp) results, arising from the studyof multiple random walks on a random regular graph G. For k independentwalks on G, the cover timeCG(k) is asymptotic to CG/k, where CG is thecover time of a single walk. For most starting positions, the expected numberof steps before any of the walks meet is θrn/(k2). If the walks can communi-cate when meeting at a vertex, we show that, for most starting positions, theexpected time for k walks to broadcast a single piece of information to eachother is asymptotic to 2 ln kk θrn, as k, n→∞.We also establish properties of walks where there are two types of parti-cles, predator and prey, or where particles interact when they meet at a vertexby coalescing, or by annihilating each other. For example, the expected ex-tinction time of k explosive particles (k even) tends to (2 ln 2)θrn as k →∞.The case of n coalescing particles, where one particle is initially locatedat each vertex, corresponds to a voter model defined as follows: Initiallyeach vertex has a distinct opinion, and at each step each vertex changes its∗Department of Computer Science, King’s College, University of London, London WC2R 2LS,UK ([email protected], [email protected]). Research supported by Royal SocietyGrant 2006/R2-IJP.†Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.Email: [email protected]. Research supported in part by NSF grant CCF0502793.