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arXiv · 2607.15522

Nonnegative Bakry--\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\'e Inequalities

Qi Guo·Xueping Huang·Yi C. Huang
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Abstract

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--\'Emery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincar\'e 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 M\"unch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincar\'e inequality.

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Qi Guo, Xueping Huang, Yi C. Huang. 2026-07-17. Nonnegative Bakry--\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\'e Inequalities. https://arxiv.org/abs/2607.15522

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