Distribution of Money on Connected Graphs with Multiple Banks

¹ School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287, USA
² Mathematics Department California State University Fullerton, CA 92834, USA

^* Corresponding author: nlanchie@asu.edu

Received: 29 November 2023
Accepted: 17 May 2024

Abstract

This paper studies an interacting particle system of interest in econophysics inspired from a model introduced in the physics literature. The original model consists of the customers of a single bank characterized by their capital, and the dynamics consists of monetary transactions in which a random individual x gives one coin to another random individual y, the transaction being canceled when x is in debt and there are no more coins in the bank. Using a combination of numerical simulations and heuristic arguments, physicists conjectured that the distribution of money (the random number of coins owned by a given individual) at equilibrium converges to an asymmetric Laplace distribution in the large population limit when the money temperature is large. We prove and extend this conjecture to a more general model including multiple banks and interactions among customers across banks. More importantly, we assume that customers are located on a general undirected connected graph (as opposed to the complete graph in the original model) where neighbors are interpreted as business partners, and transactions occur along the edges, thus modeling the flow of money across a social network. We first derive an exact expression for the distribution of money for all population sizes and money temperatures, then prove its convergence to an asymmetric Laplace distribution in the large population limit.