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

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