Fhj Frank Redig

Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.

Duality and exact correlations for a model of heat conduction

We study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits long-range correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process.

Large deviations for quantum spin systems

AbstractWe consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages

Dynamics of condensation in the symmetric inclusion process

\bar X_\Lambda : = \frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } X_i

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

, where the Xis are copies of a self-adjoint element X (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of

The Potts model built on sand

\frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } \delta _{X_i }