On coalescence time in graphs-When is coalescing as fast as meeting?

Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as Cooper et al. [14] and Berenbrink et al. [7], the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to coalesce. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log n), then the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almost-regular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n). By duality, our results give bounds on the voter model. keywords— coalescing time, meeting time, random walks, voter model ∗Supported by the ERC Starting Grant (DYNAMIC MARCH).