A note about the uniform distribution on the intersection of a simplex and a sphere

TL;DR

This paper investigates the uniform distribution on the intersection of a simplex and a sphere, revealing phase transitions and localization phenomena in high-dimensional settings.

Contribution

It introduces new insights into the behavior of uniform distributions on simplex-sphere intersections, highlighting phase transitions and localization effects.

Findings

Identification of phase transition phenomena

Observation of localization phenomena

Analysis of high-dimensional distribution behavior

Abstract

Uniform probability distributions on $\ell_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a simplex and a sphere. Certain new and interesting features, such as phase transitions and localization phenomena emerge.