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In this paper, we continue our analysis of spatial versions of agent-basedmodels for the dynamics of money that have been introduced in the statisticalphysics literature, focusing on two models with debts. Both models consist ofsystems of economical agents located on a finite connected graph representing asocial network. Each agent is characterized by the number of coins she has,which can be negative in case she is in debt, and each monetary transactionconsists in one coin moving from one agent to one of her neighbors. In thefirst model, that we name the model with individual debt limit, the agents areallowed to individually borrow up to a fixed number of coins. In the secondmodel, that we name the model with collective debt limit, agents can borrowcoins from a central bank as long as the bank is not empty, with reimbursementsoccurring each time an agent in debt receives a coin. Based on numericalsimulations of the models on complete graphs, it was conjectured that, in thelarge population/temperature limits, the distribution of money converges to ashifted exponential distribution for the model with individual debt limit, andto an asymmetric Laplace distribution for the model with collective debt limit.In this paper, we prove exact formulas for the distribution of money that arevalid for all possible social networks. Taking the large population/temperaturelimits in the formula found for the model with individual debt limit, we proveconvergence to the shifted exponential distribution, thus establishing thefirst conjecture. Simplifying the formula found for the model with collectivedebt limit is more complicated, but using a computer to plot this formula showsan almost perfect fit with the Laplace distribution, which strongly supportsthe second conjecture.

Rigorous results for the distribution of money on connected graphs (models with debts)