What happens to a random walk before equilibrium?
Seminar Room 1, Newton Institute
Since the pioneering work of Diaconis and Shahshahani, much research has been devoted to the study of the cutoff phenomenon, which describes how a random walk on a finite graph reaches its equilibrium distribution over a dramatically short period of time. However, much less is known about how a random walk behaves before it has reached this equilibrium distribution. The purpose of this talk is to study aspects of this question, with an emphasis on the way a random walk gets away from its starting point. In some interesting cases, the growth of the distance exhibits a phase transition from linear to sublinear behaviour. In other examples, there are different regimes with different scalings but no phase transition. While we do not have a theory at the moment, I will discuss some results which relate this phenomenon to the local geometry of the graph (as perceived by a random walker) and state some conjectures.
Partly joint with Rick Durrett.
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