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

Probability Seminar

13th October 2017, 3:30 pm – 4:30 pm

Main Maths Building, SM4

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. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time.

As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log^2 n), 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.

(joint work with Varun Kanade and Frederik Mallmann-Trenn)

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