Laplacians and Random Walks on CW Complexes

TL;DR

This paper introduces random walks on CW complexes and links their properties to cellular Laplacians, enabling the recovery of Novikov-Shubin invariants from return probabilities, thus connecting topological and probabilistic methods.

Contribution

It constructs new random walks on CW complexes and relates their operators to cellular Laplacians, providing a novel approach to compute Novikov-Shubin invariants.

Findings

Random walks on CW complexes are defined and analyzed.

Operators related to these walks connect to cellular Laplacians.

Novikov-Shubin invariants can be recovered from return probabilities.

Abstract

We construct random walks taking place on the k-cells of free G-CW complexes of finite type. These random walks define operators acting on the cellular k-chains that relate nicely to the (upper) cellular k-Laplacian. As an application, we use this relation to show that the Novikov-Shubin invariants of a free G-CW complex X of finite type can be recovered from quantities related to return probabilities of the random walks on the cells of X.