Nonnegative Bakry--Émery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincaré Inequalities

Abstract

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.

Disclosure

“thesis are required. □ Declarations of interest: None. Data availability statement: There are no new data associated with this article. AI assistance statement: The authors used AI tools to assist with initial conceptualization, symbolic checking and manuscript editing; all mathematical validation, final proof decisions, and final wording remain the sole responsibility of the human authors.”

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2026-08-23T18:56:15.722287Z

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