The maximum entropy principle and volumetric properties of Orlicz balls

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

Kabluchko, Zakhar¹  Prochno, Joscha² 

[1] Univ Munster, Fac Math, Orleans Ring 10, D-48149 Munster, Germany

[2] Karl Franzens Univ Graz, Inst Math & Sci Comp, Heinrichstr 36, A-8010 Graz, Austria

关键词: Central limit theorem;    Gibbs measures;    Maximum entropy principle;    Orlicz spaces;    Sharp large deviations;    Threshold phenomenon;    Volume ratio;   

DOI  :  10.1016/j.jmaa.2020.124687

来源: Elsevier

【 摘 要 】

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschlager (1991) [32] for l(p)(d)-balls. As another application, we determine the precise asymptotic volume ratio for 2-concave Orlicz spaces l(M)(d). Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the p-generalized Gaussian distribution occurs in problems related to the geometry of l(p)(d)-balls, which are Orlicz balls when the Orlicz function is M(t) = vertical bar t vertical bar(p). (C) 2020 The Author(s). Published by Elsevier Inc.

【 授权许可】

Free