  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 discrete-time 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 is no more coins to borrow from
the bank. Using a combination of numerical simulations and heuristic arguments,
physicists conjectured that the distribution of money (the distribution of the
number of coins owned by a given individual) at equilibrium converges to an
asymmetric Laplace distribution in the large population/temperature limit. In
this paper, we prove and extend this conjecture to a more general model
including multiple banks and interactions among customers across banks. More
importantly, our model assumes 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 show the convergence to the asymmetric Laplace distribution in the large
population/temperature limit for any graph, thus proving and extending the
conjecture from the physicists, and derive an exact expression of the
distribution of money for all population sizes and money temperatures.
