Interactions between Ehrenfest’s urns arising from group actions

Abstract

The Ehrenfest diffusion model is a well-known classical physical model consisting of two urns and n balls. There is a group theoretical interpretation of the model by using the Gelfand pair $$({\mathbb {Z}}/2{\mathbb {Z}}\wr S_{n},S_{n})$$ by Diaconis and Shahshahani (Z Wahrsch Verw Gebiete 57(2):159–179, 1981). This interpretation is still valid for an r-urns generalization. Then the corresponding Gelfand pair is $$(S_{r}\wr S_{n},S_{r-1}\wr S_{n})$$ . However, in these models, there are no restrictions for ball movements, i.e., each ball can freely move to any urns. In this paper, interactions between urns arising from actions of finite groups are introduced. Degree of freedom of ball movements are restricted by finite group actions. We then show that the cutoff phenomenon occurs in some particular (yet significant and interesting) cases.