Phase Transitions in Activated Random Walk

Seminar | August 23 | 3:10-4 p.m. | 1011 Evans Hall

 Riddhipratim Basu, Stanford University

 Department of Statistics

On a locally finite graph, consider the following interacting particle system, known as
Activated Random Walk (ARW). Start with a mass density µ of initially active particles, each of
which performs a continuous time nearest neighbour symmetric random walk at rate one and falls
asleep at rate λ > 0. Sleepy particles become active on coming in contact with active particles.
I shall describe two recent works on this model: one on the infinite lattice Z, the second on the
finite periodic lattice Z /n Z. On Z, Rolla and Sidoravicius (Invent. Math., 2012) recently showed
that for small enough particle density, almost surely the number of jumps at each site is finite. We
complement the Rolla-Sidoravicius result by confirming a further physics prediction establishing
that the critical density goes to zero along with the sleep rate. On the n-cycle, if the total number
of particles is no more than n, almost surely the process reaches an absorbing state. We show a
parallel quantitative phase transition by showing that the total number of jumps until absorption
scales linearly in n if µ < λ/(λ+1) and exponentially in n if λ is sufficiently small compared to µ.
Based on joint works with Shirshendu Ganguly, Christopher Hoffman, Jacob Richey.