AI 中文总结
研究有界度连通简单图，满足特定Bakry--Émery条件时的性质。通过改编图论修正非线性热流方法，证明其体积倍增且支持特定庞加莱不等式，以更强形式解决多项式增长猜想。
AI 中文摘要
我们证明，对于未归一化拉普拉斯算子，每个满足经典无维数Bakry--Émery条件$\mathrm{CD}(0,\infty)$的有界度连通简单图是体积倍增的，并且在所有整数图尺度上都支持一个膨胀因子为二的尺度不变$L^2$ - 庞加莱不等式，其常数仅取决于最大度。这以更强的形式解决了Cushing、Liu和Peyerimhoff的多项式增长猜想。主要创新点是对Münch引入并由Pajot和Russ扩展到无限加权图的图论修正非线性热流方法的无维数改编：$\Gamma_2\geq0$的点质量结果和正预解式平滑取代了任何全局$\mathrm{CD}(0,n)$约简，而扩散退出时间控制和有限体积局部化产生了庞加莱不等式。
英文摘要
We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--Émery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincaré inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by Münch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $Γ_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.
