AI 中文总结
该研究针对非负Ollivier-Ricci曲率的有界度图，证明了连续时间随机游走位移平方期望与球体积对数的逐点子指数增长上界，揭示了此类图的几何与随机游走性质。
AI 中文摘要
设$G=(V,E)$为可能无限、局部有限且非负Ollivier-Ricci曲率、度数界为$d<\f$的图。我们证明存在常数$C_d$，使得对任意顶点$x$及所有$r,t\be^e$，连续时间随机游走位移平方的期望满足$\bE_x \text{dist}(x,X_t)^2 \tle t \text{exp}\big[C_d \text{sqrt}(\text{log } t \text{ log log } t)\big]$，球体积对数满足$\text{log Vol}(B(x,r)) \tle \text{exp}\big[C_d \text{sqrt}(\text{log } r \text{ log log } r)\big]$。
英文摘要
Let $G=(V,E)$ be a possibly infinite, locally finite graph with non-negative Ollivier--Ricci curvature and degrees bounded by $d<\infty$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy \[ \mathbb{E}_x \mathrm{dist}(x,X_t)^2 \le t \exp\left[C_d \sqrt{\log t \log\log t}\right], \] \[ \log \mathrm{Vol}(B(x,r)) \le \exp\left[C_d \sqrt{\log r \log\log r}\right], \] for every $x\in V$ and all $r,t \ge e^e$.
Comments23 pages
