We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle Z /n Z. One starts with a mass density µ > 0 of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate λ > 0. Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters µ and λ. On the finite graph Z /n Z, unless there are more than n particles, the process fixates (reaches an absorbing state) almost surely in finite time. In a first rigorous result for a finite system, establishing well known beliefs in the statistical physics literature, we show that the number of steps the process takes to fixate is linear in n (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in n when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established in [22]. arXiv:1709.09163v2 [math.PR]

It is well known that an n × n Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if d is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when d = Θ(n 3 ). Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when d/n 3 → c ∈ (0, ∞). This shows, in particular, that the phase transition from Wishart to GOE is smooth. * Microsoft Research; miracz@microsoft.com.

We consider the Activated Random Walk model on Z. In this model, each particle performs a continuoustime simple symmetric random walk, and falls asleep at rate λ. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate λ < ∞ if the density ζ is close enough to 1 then the system stays active.

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