Interactions between Ehrenfest’s urns arising from group actions. (English) Zbl 1410.60014
Summary: 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 P. Diaconis and M. Shahshahani [Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 159–179 (1981; Zbl 0485.60006)]. 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.
MSC:
| 60C05 | Combinatorial probability |
| 05E05 | Symmetric functions and generalizations |
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
Keywords:
finite Gelfand pair; finite homogenous space; the Ehrenfest diffusion model; the cutoff phenomenonCitations:
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