118881 20250109150035.0 doi 10.1016/j.jfa.2021.109291 sideral 124982 ART-2022-124982 eng Alonso Gutiérrez, David Universidad de Zaragoza (orcid)0000-0003-1256-3671 Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures 2022 Access copy available to the general public Unrestricted In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $\sqrt{n}$ times the asymptotic value of $n^{-1/2}\left(\E\left[\Vert X_n\Vert_2^2\right]\right)^{1/2}$ (as $n\to\infty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {\bf 495}(1) 2021, 1--19]. info:eu-repo/grantAgreement/ES/DGA/E48-20R info:eu-repo/grantAgreement/ES/MICINN PID2019-105979GB-I00 info:eu-repo/semantics/openAccess by-nc-nd http://creativecommons.org/licenses/by-nc-nd/3.0/es/ 1.7 2022 MATHEMATICS 51 / 329 = 0.155 2022 Q1 T1 1.959 2022 Analysis 2022 Q1 2.9 2022 info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion Prochno, Joscha 2006 015 Universidad de Zaragoza Dpto. Matemáticas Área Análisis Matemático 282, 1 (2022), 109291 [35 pp.] J. funct. anal. JOURNAL OF FUNCTIONAL ANALYSIS 0022-1236 1152950 http://zaguan.unizar.es/record/118881/files/texto_completo.pdf Postprint 1720532 http://zaguan.unizar.es/record/118881/files/texto_completo.jpg?subformat=icon icon Postprint oai:zaguan.unizar.es:118881 articulos driver 2025-01-09-14:59:00 ARTICLE