期刊
AXIOMS
卷 11, 期 6, 页码 -出版社
MDPI
DOI: 10.3390/axioms11060245
关键词
Lie groups; invariant measures; concentration; topological dynamics
资金
- projects MEGABIT-Universita degli Studi di Catania, PIAno di inCEntivi per la RIcerca di Ateneo 2020/2022 (PIACERI), Linea di intervento 2
This paper studies the phenomenon of concentration of measures in families of compact connected Lie groups. It provides explicit examples for the determination of the region where the measure concentrates, using Macdonald's formula and Ricci curvature analysis.
We consider the phenomenon of concentration of measures, which is restricted to the case of families of compact connected Lie groups. While in the literature, powerful general results regarding the existence of concentration and its relations to extremal amenability of infinite dimensional groups have been determined, there are few explicit examples, specially regarding the determination of the region where the measure concentrates. Since they can be relevant for concrete applications, both in mathematics and in physics, in the present paper, we provide a number of such examples, using compact Lie groups as basic ingredients. In particular, our strategy is to employ the Macdonald's formula, giving the volume of compact simple Lie groups, and Ricci curvature of the bi-invariant metric for analyzing a concentration locus, which is a tool to detect where a sequence of metric, Borel measurable spaces concentrates its measure.
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