|
Résumé |
The Raise and Peel model is a stochastic model of fluctuating interface with
non-local avalanche dynamics.
Its dynamical rules stem from the structure of a specific representation of the
Temperley-Lieb algebra, and its stochastic generator can be represented by the
Hamiltonian of the XXZ Heisenberg quantum spin chain with twisted boundary
conditions. We describe the recent progress in description of the statistics
of the avalanches in the model. We focus on the large deviations of two
avalanche currents in the thermodynamic limit, which reveal a phase transition
in the behaviour of one current conditioned to atypical values of another one.
We also describe a proof of the laws of large numbers for the two currents at
arbitrary finite lattices, which in addition proves two earlier conjectures on
the structure of the stationary state of the model. The technique is based on
the analysis of the largest eigenvalue of the deformed stochastic generator
using the Bethe ansatz and Baxter's T-Q relation. |
