Large deviations, moderate deviations, and the KLS conjecture

Alonso Gutiérrez, David (Universidad de Zaragoza) ; Prochno, Joscha ; Thäle, Christoph

Resumen: Having its origin in theoretical computer science, the Kannan-Lov\'asz-Simonovits (KLS) conjecture is one of the major open problems in asymptotic convex geometry and high-dimensional probability theory today. In this work, we establish a connection between this conjecture and the study of large and moderate deviations for isotropic log-concave random vectors. We then study the moderate deviations for the Euclidean norm of random orthogonally projected random vectors in an $\ell_p^n$--ball. This leads to a number of interesting observations: (A) the $\ell_1^n$--ball is critical for the new approach; (B) for $p\geq 2$ the rate function in the moderate deviations principle undergoes a phase transition, depending on whether the scaling is below the square-root of the subspace dimensions or comparable; (C) for $1\leq p<2$ and comparable subspace dimensions, the rate function again displays a phase transition depending on its growth relative to n^{p/2}.

Idioma: Inglés

DOI: 10.1016/j.jfa.2020.108779

Año: 2021

Publicado en: JOURNAL OF FUNCTIONAL ANALYSIS 280, 1 (2021), 108779 1-33
ISSN: 0022-1236

Factor impacto JCR: 1.891 (2021)

Categ. JCR: MATHEMATICS rank: 47 / 333 = 0.141 (2021) - Q1 - T1

Factor impacto CITESCORE: 2.7 - Mathematics (Q2)

Factor impacto SCIMAGO: 1.819 - Analysis (Q1)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-20R
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2016-77710-P
Financiación: info:eu-repo/grantAgreement/ES/MICINN/PID2019-105979GP-I00

Tipo y forma: Artículo (PostPrint)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)