发表机构
Beijing Institute of Mathematical Sciences and Applications, and Yau Mathematical Sciences Center, Tsinghua University（北京国际数学研究中心与清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对可数网络上的弱强化波利亚瓮，证明了其归一化边权重依坐标几乎必然收敛到与初始权重无关的确定性向量，解决了相关均质化猜想。
AI 中文摘要
我们研究度数一致有界的可数图上的弱强化波利亚瓮，其顶点具有正的有界触发率，且触发率不必远离零。对于每个强化指数α∈[0,1)以及有限正初始权重的任意确定性选择，归一化边权重依坐标几乎必然收敛到与初始权重无关的确定性向量。我们将该向量确定为极限方程的唯一非零平衡，从而证明了库齐涅和赫希的均质化猜想，包括MATRIX开放问题报告中针对非均匀触发率的表述。主要步骤是极限方程的刘维尔定理，该定理通过利用每个顶点的行和平衡对完全解进行对数范数比较得以建立。
英文摘要
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 $α\in[0,1)$ 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.
