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

Abstract

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 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 Gamma(s + t) 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 selfduality is linked with an underlying SU (1, 1) symmetry, reminiscent of the one found before for the Symmetric Inclusion Process and related processes.

Generalized immediate exchange models and their symmetries

“…So we obtain the following theorem. Notice that this theorem was already proved in [10] with the help of direct somewhat tedious computations with hypergeometric functions. for α ∈ {+, −, 0}.…”

“…The immediate exchange model is a model of wealth distribution, introduced in [2], further studied in [3], generalized and studied from the viewpoint of processes with duality in [10].…”

“…We showed in [10] that the splitting can be done according to a B(s, t) distribution, and then the model has reversible product measures of type Γ(s + t). This was established using a duality with a discrete model of the same type, where discrete mass is redistributed in an analogous way, and where there the splitting part is using a Beta Binomial distribution.…”

Heat conduction and the nonequilibrium stationary states of stochastic energy exchange processes

“…The notion that the r-point correlation functions in the nonequilibrium steady state can be obtained from absorption probabilities of r dual particles has in the recent years led to a number of other fruitful applications of duality in the context of interacting particle systems [4][5][6][7][8][9].…”

“…A potent tool which has proven very useful in establishing the properties of the KMP model is that of duality [3], which allows to reduce the analysis of the model under consideration to that of a lattice gas of particles randomly hopping and mixing among neighbouring sites until they are absorbed at the boundaries; see also reference [2]. The notion that the r-point correlation functions in the nonequilibrium steady state can be obtained from absorption probabilities of r dual particles has in the recent years led to arXiv:1703.01240v1 [cond-mat.stat-mech] 3 Mar 2017 a number of other fruitful applications of duality in the context of interacting particle systems [4][5][6][7][8][9].…”