arXiv subjects

Ryunosuke Ozawa

Publications and source records attributed to Ryunosuke Ozawa.

5 recordsLinked to original sources

arXiv2022

Heat flow and concentration of measure on directed graphs with a lower Ricci curvature bound

In a previous work, the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower Ricci curvature bound via a gradient estimate for the heat semigroup, and a transportation inequality along the heat flow. As an application, we will conclude a concentration of measure inequality for directed graphs of positive Ricci curvature.

math.DG↗

arXiv2021

Maximal diameter theorem for directed graphs of positive Ricci curvature

In a previous work, the authors have introduced a Lin-Lu-Yau type Ricci curvature for directed graphs, and obtained a diameter comparison of Bonnet-Myers type. In this paper, we investigate rigidity properties for the equality case, and conclude a maximal diameter theorem of Cheng type.

math.DG↗

arXiv2020

Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization by using the mean transition probability kernel which appears in the formulation of the Chung Laplacian. We conclude several geometric and spectral properties of directed graphs under a lower Ricci curvature bound extending previous results in the undirected case.

math.DG↗

arXiv2014

Estimate of observable diameter of $l_p$-product spaces

We estimate the observable diameter of the $l_p$-product space $X^n$ of an mm-space $X$ by using the limit formula in our previous paper. The idea of our proof is based on Gromov's book. As a corollary we obtain the phase transition property of $\{X^n\}_{n=1}^\infty$ under a discreteness condition.

math.MG↗

arXiv2014

Limit formulas for metric measure invariants and phase transition property

We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition property for a sequence of metric measure spaces or pyramids, and find some examples of symmetric spaces of noncompact type with the phase transition property. We also give a simple proof of a theorem by Funano-Shioya on the limit of an $N$-Lévy family.

math.MG↗