发表机构
HUN-REN Alfréd Rényi Institute of Mathematics; Einstein Institute of Mathematics, Hebrew University of Jerusalem（匈牙利科学院阿尔弗雷德·雷尼数学研究所; 耶路撒冷希伯来大学爱因斯坦数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有限连通简单图的均匀生成树与随机1-出子图建立耦合，推广至行列式过程与1-出过程的耦合，推导了相关同调挠率的指数增长率。
AI 中文摘要
我们证明，对于每个含n个顶点、最小度k≥2的有限连通简单图G，存在其均匀生成树与随机1-出子图的耦合，使得二者不同边的期望数量为O(n log k / √k)。更一般地，我们在一大类行列式过程及其对应1-出过程（包括Kalai的行列式超图）之间建立了类似耦合，其期望对称差相对于行列式集合的大小可忽略不计。尽管图论情形有出人意料的简短简洁证明，但更一般的结果需要建立这类行列式测度的渐近等分性质，由此我们还推导了高维正则高度复形中行列式超森林的同调挠率的指数增长率。
英文摘要
We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings between a wide class of determinantal processes and their corresponding $1$-out processes, including Kalai's determinantal hypertrees, with expected symmetric difference negligible compared to the size of the determinantal set. While the graphical case admits a surprisingly short and elegant proof, the more general result requires establishing an asymptotic equipartition property for the determinantal measures in this class, from which we also derive exponential growth rates for the homology torsion of determinantal hyperforests in higher-dimensional regular high-degree complexes.
