Duality and Stationary Distributions of the “Immediate Exchange Model” and Its Generalizations

We study the “Immediate Exchange Model”, a wealth distribution model introduced in Heinsalu and Patriarca (Eur Phys J B 87:170, 2014). We prove that the model has a discrete dual, where the duality functions are natural polynomials associated to the Gamma distribution with shape parameter 2 and are exactly those connecting the Brownian Energy Process (with parameter 2) and the corresponding Symmetric Inclusion Process in Carinci et al. (J Stat Phys 152:657–697, 2013) and Giardinà et al. (J Stat Phys 135(1):25–55, 2009). As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties for a more general model, where the exchange fraction is Beta(s, t) distributed, and product measures with (Formula presented.) marginals are invariant. We also show that the discrete dual model itself is self-dual and has the original continuous model as its scaling limit. We show that the self-duality is linked with an underlying SU(1, 1) symmetry, reminiscent of the one found before for the Symmetric Inclusion Process and related processes.