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Florentin Münch

Publications and source records attributed to Florentin Münch.

At least 37 records · Page 2Linked to original sources
arXiv2023

Bakry-Émery calculus for entropic curvature, new diameter estimates, and spectral gaps

In this paper, we propose a generalization of Bakry-Émery's calculus which allows us to formulate both Bakry-Émery and entropic curvature simultaneously. This formulation represents both curvatures as an integral of the Bochner formula against some measure. This leads to a natural optimality criterion of measures, which we investigate in the Bakry-Émery setting. Moreover, our approach leads also to a dimension parameter in the framework of entropic curvature. We also present gradient estimate applications, that is, diameter estimates for Markov chains with strictly positive entropic curvature and a spectral gap estimate. The latter implies non-existence of non-negatively curved expanders.

math.DG↗
arXiv2023

Bakry-Émery and Ollivier Ricci Curvature of Cayley Graphs

In this article we study two discrete curvature notions, Bakry-Émery curvature and Ollivier Ricci curvature, on Cayley graphs. We introduce Right Angled Artin-Coxeter Hybrids (RAACHs) generalizing Right Angled Artin and Coxeter groups (RAAGs and RACGs) and derive the curvatures of Cayley graphs of certain RAACHs. Moreover, we show for general finitely presented groups $Γ= \langle S \, \mid\, R \rangle$ that addition of relators does not lead to a decrease the weighted curvatures of their Cayley graphs with adapted weighting schemes.

math.CO↗
arXiv2023

Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property

We study heat kernel convergence of induced subgraphs with Neumann boundary conditions. We first establish convergence of the resulting semigroups to the Neumann semigroup in $\ell^2$. While convergence to the Neumann semigroup always holds, convergence to the Dirichlet semigroup in $\ell^2$ turns out to be equivalent to the coincidence of the Dirichlet and Neumann semigroups while convergence in $\ell^1$ is equivalent to stochastic completeness. We then investigate the Feller property for the Neumann semigroup via generalized solutions and give applications to graphs satisfying a condition on the edges as well as birth-death chains.

math.FA↗
arXiv2023

Ollivier curvature, Isoperimetry, concentration, and Log-Sobolev inequalitiy

We introduce a Laplacian separation principle for the the eikonal equation on Markov chains. As application, we prove an isoperimetric concentration inequality for Markov chains with non-negative Ollivier curvature. That is, every single point from the concentration profile yields an estimate for every point of the isoperimetric estimate. Applying to exponential and Gaussian concentration, we obtain affirmative answers to two open quesions by Erbar and Fathi. Moreover, we prove that the modified log-Sobolev constant is at least the minimal Ollivier Ricci curvature, assuming non-negative Ollivier sectional curvature, i.e., the Ollivier Ricci curvature when replacing the $\ell_1$ by the $\ell_\infty$ Wasserstein distance. This settles a recent open Problem by Pedrotti. We give a simple example showing that non-negative Ollivier sectional curvature is necessary to obtain a modified log-Sobolev inequality via positive Ollivier Ricci bound. This provides a counterexample to a conjecture by Peres and Tetali.

math.DG↗
arXiv2022

Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. I. Theory

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-Émery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserve the Markovian property, its limits turn out to be curvature sharp. Our aim is to present the flow in the most general case of not necessarily reversible random walks allowing laziness, including vanishing transition probabilities along some edges ("degenerate" edges). This approach requires to extend all concepts (in particular, the Bakry-Émery curvature related notions) to this general case and it leads to a distinction between the underlying topology (a mixed combinatorial graph) and the weighting scheme (given by transition rates). We present various results about curvature sharp vertices and weighted graphs as well as some fundamental properties of this new curvature flow. This paper is accompanied by a second paper discussing the curvature flow implementation in Python for practical use. In this second paper we present examples and exhibit further properties of the flow.

math.CA↗
arXiv2022

Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. II. Implementation

In this second part of a sequence of two papers, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry-Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the first paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, hypercubes) and exhibit further properties of this flow. One particular aspect in our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the available Python functions and routines available in the ancillary file. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

math.CA↗
arXiv2022

Ollivier curvature, betweenness centrality and average distance

We give a new upper bound for the average graph distance in terms of the average Ollivier curvature. Here, the average Ollivier curvature is weighted with the edge betweenness centrality. Moreover, we prove that equality is attained precisely for the reflective graphs which have been classified as Cartesian products of cocktail party graphs, Johnson graphs, halved cubes, Schläfli graphs, and Gosset graphs.

math.CO↗
arXiv2022

Mixing time and expansion of non-negatively curved Markov chains

We establish three remarkable consequences of non-negative curvature for sparse Markov chains. First, their conductance decreases logarithmically with the number of states. Second, their displacement is at least diffusive until the mixing time. Third, they never exhibit the cutoff phenomenon. The first result provides a nearly sharp quantitative answer to a classical question of Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for graphs with non-negative curvature. The third offers a striking counterpoint to the recently established cutoff for non-negatively curved chains with uniform expansion.

math.PR↗
arXiv2022

Reflective Graphs, Ollivier curvature, effective diameter, and rigidity

We give a discrete Bonnet Myers type theorem for the effective diameter assuming positive Ollivier curvature. We prove that this diameter bound is attained if and only if the graph is a cocktail party graph, a Johnson graph, a halved cube, a Schläfli graph, a Gosset graph, or a cartesian product of the mentioned graphs with same Ollivier curvature. As a key step in the proof, we introduce the notion of reflective graphs as graphs such that for any two neighbors there exists a certain self-inverse automorphism mapping one neighbor to another. We classify these graphs as arbitrary cartesian products of the graphs mentioned before.

math.CO↗
arXiv2022

Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems.

math.AP↗
arXiv2021

Characterizations of Forman curvature

We characterize Forman curvature lower bounds via contractivity of the Hodge Laplacian semigroup. We prove that Ollivier and Forman curvature coincide on edges when maximizing the Forman curvature over the choice of 2-cells. To this end, we translate between 2-cells and transport plans. Moreover, we give improved diameter bounds. We explicitly warn the reader that our Forman curvature notion does not coincide with Forman's original definition, but can be seen as generalization of the latter one.

math.DG↗
arXiv2021

Every Salami has two ends

A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.

math.DG↗
arXiv2020

Spectral gap of the largest eigenvalue of the normalized graph Laplacian

We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $\frac{n-1}2$. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $\frac{n-1}{2}$.

math.SP↗
arXiv2019

Spectrally positive Bakry-Émery Ricci curvature on graphs

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.

math.DG↗
arXiv2019

Li-Yau inequality under $CD(0,n)$ on graphs

We introduce a modified non-linear heat equation $\partial_t u = Δu + Γu$ as a substitute of $\log P_t f$ where $P_t$ is the heat semigroup. We prove an exponential decay of $Γu$ under the Bakry Emery curvature condition $CD(K,\infty)$ and prove the Li-Yau inequality $-Δu_t \leq \frac{n}{2t}$ under the Bakry Emery curvature condition $CD(0,n)$. From this, we deduce the volume doubling property which solves a major open problem in discrete Ricci curvature. As an application, we show that there exist no expander graphs satisfying $CD(0,n)$.

math.DG↗
arXiv2019

Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $λ_1 \geq \log(2)/\operatorname{diam}^2$.

math.DG↗