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

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.
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  • HAL Id : ensl-00608829, version 2
  • ARXIV : 1107.3227

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Cedric Bernardin, Fabio Lucio Toninelli. A one-dimensional coagulation-fragmentation process with a dynamical phase transition. Stochastic Processes and their Applications, Elsevier, 2012, 122 (4), pp.35. ⟨ensl-00608829v2⟩

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