A one-dimensional coagulation-fragmentation process with a dynamical phase transition

TL;DR

This paper studies a reversible coagulation-fragmentation process on partitions of a set, revealing a dynamical phase transition with distinct equilibrium behaviors and scaling of mixing times depending on the phase.

Contribution

It introduces a new Markovian coagulation-fragmentation model with a phase transition and analyzes the different equilibrium regimes and their mixing times.

Findings

Existence of three phases: delocalized, localized, and critical.

Different scaling of mixing times in each phase.

Propagation of fragmentation fronts explains equilibration in the localized phase.

Abstract

We introduce a reversible Markovian coagulation-fragmentation process on the set of partitions of $\{1,\ldots,L\}$ into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model \cite{cf:GBbook}. Depending on a parameter $\lambda$, the typical configuration can be either dominated by a single big interval (delocalized phase), or be composed of many intervals of order $1$ (localized phase), or the interval length can have a power law distribution (critical regime). In the three cases, the time required to approach equilibrium (in total variation) scales very differently with $L$. In the localized phase, when the initial condition is a single interval of size $L$, the equilibration mechanism is due to the propagation of two "fragmentation fronts" which start from the two boundaries and proceed by power-law jumps.