Discrete Curvature and Abelian Groups

By Bo'az Klartag, Gady Kozma, Peter Ralli, Prasad Tetali. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2015

89 citations recorded by the source corpus.

Abstract We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called ${{\Gamma }_{2}}$ -calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7b2cb71d1bc7f1b2f7a69ff3b6e6f8564842b242

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Bakry–Émery curvature and diameter bounds on graphs (2016, methodology): “In this article, we focus on Bakry-Émery-Ricci-curvature [1, 21, 16, 13].”
Bakry–Émery Curvature Functions on Graphs (2016, background): “(b) All abelian Cayley graphs lie in the class CD(0,∞) ([11, 25, 21]).”
Cutoff for non-negatively curved Markov chains (2021, background): “Classical examples include: • all random walks on abelian groups [42]; • all conjugacy-invariant random walks on the permutation group [26, 10]; • simple random walks on all Bruhat graphs of Coxeter groups [60]; • all monotone birth-and-death chains [40]; • the zero-range process with non-decreasing”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, methodology): “[26] studies several examples of Markov chains for which Bakry–Émery curvature bounds can be proven, such as the random transposition model and the Bernoulli–Laplace process.”
Nilprogressions and groups with moderate growth (2015, background): “hich is to say that λ1 is comparable to h2. This follows from a simple argument using the doubling property at a scale comparable to the diameter. See Lemma 5.2. It is known by work of Klartag et al. [33] that for abelian G we may take C1 = 16, which is essentially sharp. NILPROGRESSIONS AND GROUPS ”