Interactions between Ehrenfest's urns arising from group actions

TL;DR

This paper explores how group actions influence interactions in Ehrenfest's urn models, restricting ball movements and analyzing the emergence of cutoff phenomena, extending classical models with a group-theoretic perspective.

Contribution

It introduces group action-based interactions in Ehrenfest's urn models and demonstrates the existence of cutoff phenomena in these generalized settings.

Findings

Group actions restrict ball movements in urn models.

Cutoff phenomena are observed under certain group actions.

Extension of classical Ehrenfest models with group-theoretic interactions.

Abstract

Ehrenfest's diffusion model is a well-known classical physical model consisting of two urns and n balls. A group theoretical interpretation of the model by using the Gelfand pair (Z/2Zwr S_{n},S_{n}) is provided by Diaconis-Shahshahani. This interpretation remains valid for an r-urns generalization, in which case, the corresponding Gelfand pair is (S_{r}wr S_{n},S_{r-1}wr S_{n}). In these models, there are no restrictions for ball movements, i.e., each balls can freely move to any urn. This paper introduces interactions between urns arising from actions of finite groups. The degree of freedom of ball movements is restricted by finite groups actions. Furthermore, for some cases, the existence of the cut-off phenomenons is shown.