Duality and stationary distributions of the "Immediate Exchange Model" and its generalizations

Abstract

We prove that the "Immediate Exchange Model" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and prove full SU(1,1) for both the continuous and discrete model.

24 pages, one figure. Substantial reworking on the previous version; proof of self-duality for the general case, full SU(1,1) of the continuous and discrete model

References in corpus (3)

• Duality and hidden symmetries in interacting particle systems
• Kinetic models of immediate exchange
• The Immediate Exchange model: an analytical investigation

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• Spectral gap of the symmetric inclusion process
• Generalized immediate exchange models and their symmetries
• Generic power laws in higher-dimensional lattice models with multidirectional hopping