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# The Coherence Phase Transition

## April 27, 2012 @ 11:00 am

Semen Shlosman (CNRS, France and Institute for Information Transmission Problems, Russia)

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We study particle systems corresponding to highly connected queueing networks. We examine the validity of the so-called Poisson Hypothesis (PH), which predicts that the Markov process, describing the evolution of such particle system, started from a reasonable initial state, approaches the equilibrium in time independent of the size of the network.

This is indeed the case in many situations. However, there are networks for which the relaxation process slows down. This behavior reflects the fact that the corresponding infinite system undergoes a phase transition. It is characterized by the property that different nodes of the network start to evolve in a synchronous way.

Such transition can happen only when the load per node exceeds some critical value, while in the low load situation the PH behavior holds. The load thus plays here the same role as the inverse temperature in statistical mechanics.

We will mention a related open problem of ergodicity of interacting particle systems with unique stationary state.

The talk is based on joint works with A. Rybko and A. Vladimirov.