Graph Hörmander Systems

2020/06/25 by Haojian Li, Marius Junge, Li, Haojian +3 · 1 citation
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Graph theory and applications #Mathematical Physics (math-ph) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2006.14578

openalex publication_date 2020/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper extends the Bakry-Émery theorem connecting the Ricci curvature and log-Sobolev inequalities to the matrix-valued setting. Using tools from noncommuative geometry, it is shown that for a right invariant second order differential operator on a compact Lie group, a lower bound for a matrix-valued modified log-Sobolev inequality is equivalent to a uniform lower bound for all finite dimensional representations. Using combinatorial tools, we obtain computable lower bounds for matrix-valued log-Sobolev inequalities of graph-Hörmander systems using combinatorial methods.

Citations

• On the Boltzmann Equation for Weakly Nonlinear Wave Equations
• On the shortest spanning subtree of a graph and the traveling salesman problem
• Logarithmic Sobolev inequalities for finite Markov chains
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Cited by

• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

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• A counterexample to a conjecture by Salez and Youssef
• Sharp embedding results and geometric inequalities for Hörmander vector fields
• Bakry-Émery curvature on graphs as an eigenvalue problem
• Complete Logarithmic Sobolev Inequalities via Ricci Curvature Bounded Below II
• Spectrally positive Bakry-Émery Ricci curvature on graphs