Liouville property and non-negative Ollivier curvature on graphs

TL;DR

The paper proves that graphs with non-negative Ollivier curvature satisfy the Liouville property, namely every bounded harmonic function is constant, under standard uniform bounds on degrees and jump rates. It develops a transport-planned framework to connect curvature to Lipschitz/Harmonic behavior and provides explicit infinite-graph examples—zero-range processes on the line and lattices with convex potentials—that exhibit non-negative Ollivier curvature and hence Liouville. For positive curvature, it improves concentration results by establishing a Gaussian tail bound $m(f>r) \le e^{-K r^2}$ under mild assumptions, using a semigroup/Γ-calculus approach inspired by Schmuckenschläger. Together, these results deepen understanding of how discrete Ollivier curvature governs analytic properties and measure concentration on graphs, highlighting both the potential and limits of curvature-based methods in discrete settings.

Abstract

For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.

Figures (1)

• Figure 1: The upper row shows the non-normalized Bakry-Emery curvature. The lower row shows the non-normalized Ollivier curvature from munch2017ollivier. The curvature was calculated by the graph curvature calculator cushing2017graph.

Theorems & Definitions (17)

• Proposition 1.1: See munch2017ollivier
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:Liouville']}
• Theorem 3.1
• proof
• Corollary 3.2