Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries

TL;DR

This paper extends the study of random walks from permutations and hyperplane arrangements to complex arrangements and cell complexes, revealing new Markov chain properties and connections to greedoids and libraries.

Contribution

It introduces a new class of random walks on complex hyperplane arrangements and cell complexes, generalizing known models and analyzing their spectral properties.

Findings

Eigenvalues of transition matrices are non-negative and explicitly combinatorial

Walks on libraries with multiple shelves are modeled

Interval greedoids lead to similar random walks

Abstract

The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail.