Random Walks and Mixed Volumes of Hypersimplices

TL;DR

This paper introduces a method linking mixed volume calculations of certain polytopes to symmetric random walks, with a focus on hypersimplices and permutahedra, providing new insights into their geometric properties.

Contribution

It presents a novel approach connecting mixed volume computation to symmetric random walks, specifically applied to hypersimplices and permutahedra.

Findings

Established a relationship between mixed volumes and symmetric random walks.

Applied the method to hypersimplices and permutahedra.

Provided new geometric insights into these polytopes.

Abstract

Below is a method for relating a mixed volume computation for polytopes sharing many facet directions to a symmetric random walk. The example of permutahedra and particularly hypersimplices is expanded upon.