Concentration of measure on spheres and related manifolds

Abstract

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices. A further branch addresses higher order concentration (i.\,e., concentration for non-Lipschitz functions which have bounded derivatives of higher order) for ${\ell }_p^n$-spheres. This is based on a type of generalized log-Sobolev inqualities referred to as ${\mathrm{LS}}_q$-inequalities. More generally, we prove higher order concentration bounds for probability measures on ${\mathbb{R}}^n$ which satisfy an ${\mathrm{LS}}_q$-inequality. Finally, we derive concentration bounds for sequences of smooth symmetric functions on the Euclidean sphere which are closely related to Edgeworth-type expansions.