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Title: Activated Random Walk: the line between fast and slow sleep.

Abstract: Activated random walk is a process with many chips moving on a graph, which is intended to be a better-behaved version of the abelian sandpile. Chips that are awake do a simple random walk with rate 1 in continuous time. If a chip is by itself on a vertex, it falls asleep with some rate, the parameter lambda. But, if a second chip enters that vertex later, both chips wake back up and continue walking. On a finite graph, eventually all chips will fall asleep, but that may take a very long time. There is a phase transition from fast to slow sleep. I'll talk about the phase transition in the extreme case where a cycle has one chip per vertex, and a recent result by Forien and Gaudilli猫re proving that sleep is slow on the torus for any positive density of particles if lambda is small enough.

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