Random Walk of a Cat in a Building

TL;DR

This paper introduces a new probabilistic model for a cat moving through an apartment building, analyzing the determinants of associated matrices and providing examples involving complex graph structures.

Contribution

It presents a novel probabilistic model for movement in interconnected apartments and reveals a factorization of key matrices related to this model.

Findings

Determinants of stochastic and exponential distance matrices have a nice factorization.

Provides examples involving indirectly acyclic digraphs and hyperplane arrangements.

Introduces a new model for movement in interconnected spaces.

Abstract

One usually thinks of a cat moving from one room to another in an apartment as random walk model. Imagine now that it also has the possibility to go from one apartment to another by crossing some corridors. That yields a new probabilistic model for which each corridor connects the entrance rooms of several apartments. This article shows that the determinants of the stochastic and the exponential distance matrices of that model have a nice factorization. Two examples involving indirectly acyclic digraphs and hyperplane arrangements are provided.