Homogenization of weakly reinforced Pólya urns on countable networks
Abstract
We consider weakly reinforced Pólya urns on a countable graph of uniformly bounded degree. The vertices have positive bounded firing rates, which need not be bounded away from zero. For every reinforcement exponent α∈[0,1) and every deterministic choice of finite positive initial weights, the normalized edge weights converge almost surely, coordinate-wise, to a deterministic vector independent of the initial weights. We identify this vector as the unique non-vanishing equilibrium of the limiting equation. This proves the homogenization conjecture of Couzinié and Hirsch, including its formulation for non-uniform firing rates in the MATRIX open-problems report. The main step is a Liouville theorem for the limiting equation, established by a logarithmic-norm comparison of complete solutions using the row-sum balance at each vertex.
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BibTeX
@misc{qin2026homogenization,
title = {{Homogenization of weakly reinforced Pólya urns on countable networks}},
author = {Shuo Qin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11780},
eprint = {2610.11780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11780},
}
RIS
TY - UNPB
TI - Homogenization of weakly reinforced Pólya urns on countable networks
AU - Shuo Qin
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11780
UR - https://arxiv.org/abs/2610.11780
AB - We consider weakly reinforced Pólya urns on a countable graph of uniformly bounded degree. The vertices have positive bounded firing rates, which need not be bounded away from zero. For every reinforcement exponent α∈[0,1) and every deterministic choice of finite positive initial weights, the normalized edge weights converge almost surely, coordinate-wise, to a deterministic vector independent of the initial weights. We identify this vector as the unique non-vanishing equilibrium of the limiting equation. This proves the homogenization conjecture of Couzinié and Hirsch, including its formulation for non-uniform firing rates in the MATRIX open-problems report. The main step is a Liouville theorem for the limiting equation, established by a logarithmic-norm comparison of complete solutions using the row-sum balance at each vertex.
ER -