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

2026/07/17 by Qi Guo, Xueping Huang, Yi C. Huang
#math.DG #math.CO #math.PR

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Abstract

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--Émery condition CD(0,∞) for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant L²-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 Γ₂≥0 and positive-resolvent smoothing replace any global CD(0,n) reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.

Citations

• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Ricci curvature of Markov chains on metric spaces
• Ricci curvature for metric-measure spaces via optimal transport
• On the parabolic kernel of the Schrödinger operator

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