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Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes

Claudio Landim, Diego Marcondes, Insuk Seo

Abstract

We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics which do not satisfy the mixing conditions required in Beltrán and Landim (2010,2012) or in Landim et. al. (2020). As an application, we study the metastable behavior of critical zero-range processes. Let $r: S\times S\to \bb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to the uniform measure. For $α>0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^α$, $k\ge 2$. Consider a zero-range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y)$. For $α\ge 1$, in the stationary state, as the total number of particles, represented by $N$, tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if $α\ge 1$, we call the case $α=1$ critical. By applying the general method established in the first part of the article to the critical case, we show that the site which concentrates almost all particles evolves in the time-scale $N^2 \log N$ as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.

Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes

Abstract

We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics which do not satisfy the mixing conditions required in Beltrán and Landim (2010,2012) or in Landim et. al. (2020). As an application, we study the metastable behavior of critical zero-range processes. Let be the jump rates of an irreducible random walk on a finite set , reversible with respect to the uniform measure. For , let be given by , , , . Consider a zero-range process on in which a particle jumps from a site , occupied by particles, to a site at rate . For , in the stationary state, as the total number of particles, represented by , tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if , we call the case critical. By applying the general method established in the first part of the article to the critical case, we show that the site which concentrates almost all particles evolves in the time-scale as a random walk on whose transition rates are proportional to the capacities of the underlying random walk.

Paper Structure

This paper contains 38 sections, 74 theorems, 498 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Metastability of weakly mixing Markov chains
  3. Condensing zero-range processes
  4. Condensation and metastable behavior in the super-critical case
  5. Critical zero-range processes
  6. Proof of Theorem \ref{['mt0']}
  7. The conditions (C2) and (C3)
  8. Outline of the proof of Theorem \ref{['t27']}
  9. Condensation of Critical Zero-range Process
  10. Proof of Condition (C2)
  11. Upper bound of the capacity
  12. The enlarged process
  13. Visiting points
  14. Local Spectral Gap
  15. Restricted process
  16. ...and 23 more sections

Key Result

Theorem 2.2

Assume that conditions (C1) -- (C3) are in force for some $x_0 \in S$. Then, the process $\xi_N(\cdot)$ is $(\nu_N, \{{\mathcal{E}}^x_N : x\in S\}, \delta_{x_0}, L)$-metastable in the sense of Definition def1.

Figures (2)

  • Figure 1: An illustration of sets introduced in Section \ref{['sec64']} when $S=\{x,\,y,\,z\}$. We can notice from this figure that the sets ${\mathcal{J}}^{x,\,y}$, ${\mathcal{J}}^{y,\,z}$, and ${\mathcal{J}}^{z,\,x}$ are disjoint.
  • Figure 2: An illustration of the drift of the diffusion which approximates the zero-range dynamics when $N$ is large in the case where $S=\{1,\,2,\,3\}$. The red curve represents the manifold ${\mathcal{M}}_A$.

Theorems & Definitions (157)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1: Condensation in the super-critical case
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 147 more