In this chapter we consider additional models with duality properties. We start with a class of models of mass transport, both discrete and continuous, inspired by the well-known KMP (Kipnis-Marchioro-Presutti) model. We obtain these models from a “thermalization procedure” applied to the basic models of the previous chapters. The thermalization of a model preserves symmetries and duality properties. As a consequence, all these thermalized models automatically satisfy duality properties. This yields a one parameter family of discrete and continuous models of KMP type with duality properties. Other models which we obtain via thermalization include the Kac model and the Aldous averaging model. Next we study two additional models and their duality properties using the algebraic approach for the Heisenberg algebra. The first is the Ginzburg-Landau model with quadratic potential which we show to be dual to independent random walkers. The second is the Wright-Fisher diffusion with mutation (and its finite population companion the Moran model), where we have the well-known dualities of population genetics, namely duality with the coalescent. In both cases the dualities can be understood from a change of representation in the Heisenberg algebra.

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Duality for Other Models

  • Cristian Giardinà,
  • Frank Redig

摘要

In this chapter we consider additional models with duality properties. We start with a class of models of mass transport, both discrete and continuous, inspired by the well-known KMP (Kipnis-Marchioro-Presutti) model. We obtain these models from a “thermalization procedure” applied to the basic models of the previous chapters. The thermalization of a model preserves symmetries and duality properties. As a consequence, all these thermalized models automatically satisfy duality properties. This yields a one parameter family of discrete and continuous models of KMP type with duality properties. Other models which we obtain via thermalization include the Kac model and the Aldous averaging model. Next we study two additional models and their duality properties using the algebraic approach for the Heisenberg algebra. The first is the Ginzburg-Landau model with quadratic potential which we show to be dual to independent random walkers. The second is the Wright-Fisher diffusion with mutation (and its finite population companion the Moran model), where we have the well-known dualities of population genetics, namely duality with the coalescent. In both cases the dualities can be understood from a change of representation in the Heisenberg algebra.