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This paper studies an interacting particle system of interest in econophysicsinspired from a model introduced in the physics literature. The original modelconsists of the customers of a single bank characterized by their capital, andthe discrete-time dynamics consists of monetary transactions in which a randomindividual $x$ gives one coin to another random individual $y$, the transactionbeing canceled when $x$ is in debt and there is no more coins to borrow fromthe bank. Using a combination of numerical simulations and heuristic arguments,physicists conjectured that the distribution of money (the distribution of thenumber of coins owned by a given individual) at equilibrium converges to anasymmetric Laplace distribution in the large population/temperature limit. Inthis paper, we prove and extend this conjecture to a more general modelincluding multiple banks and interactions among customers across banks. Moreimportantly, our model assumes that customers are located on a generalundirected connected graph (as opposed to the complete graph in the originalmodel) where neighbors are interpreted as business partners, and transactionsoccur along the edges, thus modeling the flow of money across a social network.We show the convergence to the asymmetric Laplace distribution in the largepopulation/temperature limit for any graph, thus proving and extending theconjecture from the physicists, and derive an exact expression of thedistribution of money for all population sizes and money temperatures.

Distribution of money on connected graphs with multiple banks