  Models in econophysics, i.e., the emerging field of statistical physics that
applies the main concepts of traditional physics to economics, typically
consist of large systems of economic agents who are characterized by the amount
of money they have. In the simplest model, at each time step, one agent gives
one dollar to another agent, with both agents being chosen independently and
uniformly at random from the system. Numerical simulations of this model
suggest that, at least when the number of agents and the average amount of
money per agent are large, the distribution of money converges to an
exponential distribution reminiscent of the Boltzmann-Gibbs distribution of
energy in physics. The main objective of this paper is to give a rigorous proof
of this result and show that the convergence to the exponential distribution is
universal in the sense that it holds more generally when the economic agents
are located on the vertices of a connected graph and interact locally with
their neighbors rather than globally with all the other agents. We also study a
closely related model where, at each time step, agents buy with a probability
proportional to the amount of money they have, and prove that in this case the
limiting distribution of money is Poissonian.
