Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature
Abstract
We consider irreducible reversible Markov chains whose nonzero off-diagonal transition probabilities are bounded below by ε>0 and whose reversible measure satisfies m-≤ m(x)≤κm-. Under the Bakry--Émery curvature condition CD(K,∞), with K∈ℝ, we establish Wang's log-Harnack inequality Pt(log f)(x)≤log Ptf(y) +Cdist(x,y)2/(∫0t e2Ksd s), for bounded positive f bounded away from zero and all x,y∈ V, where C is a constant dependent only on ε. Building upon this inequality, we further prove volume doubling property: let K-:=max{−K,0} Vol(B(o,2R))≤ C0exp (C1√(K-) R) Vol(B(o,R)), R>0, where C0,C1 depend only on ε,κ. If d is the maximum combinatorial degree of the support graph, we further prove under CD(0,∞) the sharp volume growth bound Vol(B(o,R))≤ m-CR⌊ d/2⌋ for all o and R≥ 1.
Related
- Gradient estimates and volume doubling for locally finite weighted graphs with $CDψ(n,-K)$ condition
- Jacobian estimates and geometric inequalities under intermediate Ricci curvature
- Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature
- Geometric C1-stability of geodesic spheres under volume-preserving mean curvature flow
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
BibTeX
@misc{ding2026volume,
title = {{Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature}},
author = {Yanlong Ding and Shiping Liu and Chiyu Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02010},
eprint = {2611.02010},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02010},
}
RIS
TY - UNPB
TI - Volume Doubling and Geometric Inequalities on Graphs with Bakry-Émery Curvature
AU - Yanlong Ding
AU - Shiping Liu
AU - Chiyu Zhou
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02010
UR - https://arxiv.org/abs/2611.02010
AB - We consider irreducible reversible Markov chains whose nonzero off-diagonal transition probabilities are bounded below by ε>0 and whose reversible measure satisfies m_-≤ m(x)≤κm_-. Under the Bakry--Émery curvature condition CD(K,∞), with K∈ℝ, we establish Wang's log-Harnack inequality P_t(log f)(x)≤log P_tf(y) +Cdist(x,y)^2/(∫_0^t e^2Ksd s), for bounded positive f bounded away from zero and all x,y∈ V, where C is a constant dependent only on ε. Building upon this inequality, we further prove volume doubling property: let K_-:=max{−K,0} Vol(B(o,2R))≤ C_0exp (C_1√(K_-) R) Vol(B(o,R)), R>0, where C_0,C_1 depend only on ε,κ. If d is the maximum combinatorial degree of the support graph, we further prove under CD(0,∞) the sharp volume growth bound Vol(B(o,R))≤ m_-CR^⌊ d/2⌋ for all o and R≥ 1.
ER -