Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt
Abstract
Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with ℤ-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.
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BibTeX
@misc{sakagawa2026convergence,
title = {{Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt}},
author = {Hironobu Sakagawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00901},
eprint = {2610.00901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00901},
}
RIS
TY - UNPB
TI - Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt
AU - Hironobu Sakagawa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00901
UR - https://arxiv.org/abs/2610.00901
AB - Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with ℤ-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.
ER -