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Florentin Münch

Publications and source records attributed to Florentin Münch.

At least 19 recordsLinked to original sources
arXiv2026

Transitive graphs of non-negative Ollivier-Ricci curvature have polynomial growth

We prove that every transitive graph of non-negative Ollivier-Ricci curvature has polynomial growth. We deduce from this and a theorem of Brena and Brue that a finitely generated group admits a Cayley graph of non-negative Ollivier-Ricci curvature if and only if it is virtually abelian. Beyond the transitive setting, we also prove a quasi-polynomial $r^{O(\sqrt{\log r})}$ volume bound for balls around typical vertices in finite bounded-degree graphs of non-negative Ollivier-Ricci curvature, greatly strengthening a theorem of Salez (GAFA 2022).

math.DG↗
arXiv2026

Characterization of foliations via disintegration maps

In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to determine when such conditional measures arise from a metric measure foliation. Additionally, we provide a example demonstrating how this framework can be applied to study perturbations of disintegration-induced foliations.

math.MG↗
arXiv2026

On controllability, observability and stabilizability of the heat equation on discrete graphs

We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, where $H$ is the weighted Laplacian on a discrete graph $(X,b,m)$, and where $D \subseteq X$ is relatively dense. We show cost-uniform $α$-controllability by means of a weak observability estimate for the corresponding dual observation problem. We discuss optimality of our result as well as consequences on stabilizability properties.

math.OC↗
arXiv2026

The convergence and uniqueness of a discrete-time nonlinear Markov chain

In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-ακ_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.

math.DS↗
arXiv2025

On a magneto-spectral invariant on finite graphs

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-Émery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.

math.SP↗
arXiv2025

A generalized Cheeger inequality and the Steklov Problem on finite graphs

We prove generalized Cheeger inequalities for eigenvalues of Laplacians for reversible Markov chains. Then we apply Hassannezhad and Miclo's convergence result to obtain Jammes Cheeger inequalities for Steklov eigenvalues. In particular, we get a sharp estimate for the first non-trivial Steklov eigenvalue via Escobar Cheeger constant. At the end, we extend Hassannezhad and Miclo's convergence result to non-reversible Markov chains via a different method based on resolvent convergence, answering one of their questions.

math.DG↗
arXiv2025

Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces

For a bounded Lipschitz domain $Σ$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$μ_{3-β_1} \leq λ_{1},$$ where $μ_k$ ($λ_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $Σ$ and $β_1$ is the first Betti number of $Σ.$ If $Σ$ is smooth and simply connected, we can further derive the strict inequality $ μ_{3}< λ_{1}. $ This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on $1$-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets.

math.DG↗
arXiv2025

A counterexample to a conjecture by Salez and Youssef

Remarkable progress has been made in recent years to establish log-Sobolev type inequalities under the assumption of discrete Ricci curvature bounds. More specfically, Salez and Youssef have proven that the log-Sobolev constant can be lower bounded by the Bakry Emery curvature lower bound divided by the logarithm of the sparsity parameter. They conjectured that the same holds true when replacing Bakry Emery by Ollivier curvature which is often times easier to compute in practice. In this paper, we show that this conjecture is wrong by giving a counter example on birth death chains of increasing length.

math.DG↗
arXiv2025

Entropic curvature not comparable to other curvatures -- or is it?

In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.

math.DG↗
arXiv2025

Betti number estimates for non-negatively curved graphs

In this paper, we establish Betti number estimates for graphs with non-negative Ollivier curvature, and for graphs with non-negative Bakry-Émery curvature, providing a discrete analogue of a classical result by Bochner for manifolds. Specifically, we show that for graphs with non-negative Ollivier curvature, the first Betti number is bounded above by half of the minimum combinatorial vertex degree. In contrast, for graphs with non-negative Bakry-Émery curvature, we prove that the first Betti number of the path homology is bounded above by the minimum combinatorial vertex degree minus one. We further present various rigidity results, characterizing graphs that attain the upper bound on the first Betti number under non-negative Ollivier curvature. Remarkably, these graphs are precisely the discrete tori, similar to the Riemannian setting. Furthermore, we show that the results obtained using the Ollivier curvature extend to the setting of potentially non-reversible Markov chains. Additionally, we explore rigidity cases depending on the idleness parameter of the Ollivier curvature, i.e., we characterize rigidity for bone-idle graphs with non-negative Ollivier curvature that attain the upper Betti number bound. We further establish an upper bound on the first Betti number under a more general assumption, where non-negative Ollivier curvature is required only outside a finite subset. Finally, we provide several examples, e.g., we prove that for a potentially non-reversible Markov chain on a cycle of length at least five, there always exists a unique path metric with constant Ollivier curvature. Moreover, this metric has non-negative Ollivier curvature, and the upper Betti number bound is attained if and only if the curvature is zero.

math.CO↗
arXiv2024

Cheeger type inequalities associated with isocapacitary constants on graphs

In this paper, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the spectrum of the Laplace operator and the Dirichlet-to-Neumann operator for an infinite subgraph. Estimates for higher-order Steklov eigenvalues on a finite or infinite subgraph are also proved.

math.DG↗
arXiv2024

A note on Steinerberger's curvature for graphs

In this note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, and show that self-centered Bonnet-Myers sharp graphs are precisely those which are antipodal. We also discuss similarities and differences between Steinerberger and Ollivier Ricci curvature results.

math.CO↗
arXiv2024

Some variants of discrete positive mass theorems on graphs

Inspired by asymptotically flat manifolds, we introduce the concept of asymptotically flat graphs and define the discrete ADM mass on them. We formulate the discrete positive mass conjecture based on the scalar curvature in the sense of Ollivier curvature, and prove the positive mass theorem for asymptotically flat graphs that are combinatorially isomorphic to grid graphs. As a corollary, the discrete torus does not admit positive scalar curvature. We prove a weaker version of the positive mass conjecture: an asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph. Hence the combinatorial structure of an asymptotically flat graph is determined by the curvature condition, which is a discrete analog of the rigidity part for the positive mass theorem. The key tool for the proof is the discrete harmonic function of linear growth associated with the salami structure.

math.DG↗
arXiv2024

Entropy and curvature: beyond the Peres-Tetali conjecture

We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, Gibbs samplers for high-temperature spin systems, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question.

math.PR↗
arXiv2024

Bakry-Émery-Ricci curvature: An alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. There exist various types of such Ricci curvatures, capturing different aspects of network geometry. In the present work, we investigate Bakry-Émery-Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Random and small-world networks exhibit a narrow curvature distribution whereas other classes and most of the real-world networks possess a wide curvature distribution. When we compare Bakry-Émery-Ricci curvature with two other discrete notions of Ricci-curvature, Forman-Ricci and Ollivier-Ricci curvature for both model and real-world networks, we observe a high positive correlation between Bakry-Émery-Ricci and both Forman-Ricci and Ollivier-Ricci curvature, and in particular with the augmented version of Forman-Ricci curvature. Bakry-Émery-Ricci curvature also exhibits a high negative correlation with the vertex centrality measure and degree for most of the model and real-world networks. However, it does not correlate with the clustering coefficient. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. The computational time for Bakry-Émery-Ricci curvature is shorter than that required for Ollivier-Ricci curvature but higher than for Augmented Forman-Ricci curvature.

physics.comp-ph↗
arXiv2024

Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

One of the main obstacles regarding Barky Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry-Emery curvature everywhere but on the exception set.

math.DG↗
arXiv2024

Intertwining Curvature Bounds for Graphs and Quantum Markov Semigroups

Based on earlier work by Carlen-Maas and the second- and third-named author, we introduce the notion of intertwining curvature lower bounds for graphs and quantum Markov semigroups. This curvature notion is stronger than both Bakry-Émery and entropic Ricci curvature, while also computationally simpler than the latter. We verify intertwining curvature bounds in a number of examples, including finite weighted graphs and graphs with Laplacians admitting nice mapping representations, as well as generalized dephasing semigroups and quantum Markov semigroups whose generators are formed by commuting jump operators. By improving on the best-known bounds for entropic curvature of depolarizing semigroups, we demonstrate that there can be a gap between the optimal intertwining and entropic curvature bound. In the case of qubits, this improved entropic curvature bound implies the modified logarithmic Sobolev inequality with optimal constant.

math.FA↗