Generalized crested products of Markov chains

We define a finite Markov chain, called generalized crested product, which naturally appears as a generalization of the first crested product of Markov chains. A complete spectral analysis is developed and the $k$-step transition probability is given. It is important to remark that this Markov chain describes a more general version of the classical Ehrenfest diffusion model. As a particular case, one gets a generalization of the classical Insect Markov chain defined on the ultrametric space. Finally, an interpretation in terms of representation group theory is given, by showing the correspondence between the spectral decomposition of the generalized crested product and the Gelfand pairs associated with the generalized wreath product of permutation groups.Comment: 18 page

Weighted spanning trees on some self-similar graphs

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.Comment: 21 page

Schreier graphs of the Basilica group

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $\Gamma_{\xi}$, $\xi\in \partial T$. Denoting by $\xi_n$ the prefix of length $n$ of the infinite ray $\xi$, the rooted graph $(\Gamma_{\xi},\xi)$ is then the limit of the sequence of finite rooted graphs $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $\xi$.Comment: 32 page