Student Probability/PDE Seminar: Metastability of the Zero Range Process on a Finite Set Without Capacity Estimates

Seminar | April 6 | 2:10-3:30 p.m. | 891 Evans Hall

 Chanwoo Oh, UC Berkeley

 Department of Mathematics

In this talk, I'll prove metastability of the zero range process on a finite set without using capacity estimates. The proof is based on the existence of certain auxiliary functions. One such function is inspired by Evans and Tabrizian's article, "Asymptotics for the Kramers-Smouchowski equations". This function is the solution of a certain equation involving the infinitesimal generator of the zero range process. Another relevant auxiliary function is from a work of Beltran and Landim. We also use martingale problems to characterize Markov processes. Let $p$ be the jump rates of a random walk on a finite set $S$. Assume that the uniform measure on $S$ is an invariant measure of this random walk for simplicity (we expect that our method is applicable for an arbitrary invariant measure $m$). Consider the zero range process on $S$, where the rate the particle jumps from a site $x$ to $y$ with $k$ particles at the site $x$ is given by $g(k) p(x, y).$ Here $g(0) = 0$, $g(1) = 1$, and $g(k) = (k/ k-1)^\alpha , k > 1$ for some $\alpha > 1$. As total number of particles $N \rightarrow \infty $, most of the particles concentrate on a single site. In the time scale $N^{1+\alpha }$, the site of concentration evolves as a Markov chain whose jump rates are proportional to the capacities of the underlying random walk. This talk based on the joint work with F. Rezakhanlou.