Counting dimer coverings on self-similar Schreier graphs

Abstract

We study partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions of finitely generated groups by automorphisms on rooted trees, and thus their edges are naturally labeled by the generators of the group. It is thus natural to consider weight functions on these graphs taking different values according to the labeling. We study in detail the well-known example of the Hanoi Towers group $H^{(3)}$, closely related to the Sierpiński gasket.

29 pages. Final version, to appear in European Journal of Combinatorics

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• Statistics of close-packed dimers on fractal lattices
• Computing the Wiener index in Sierpinski carpet graphs
• Horofunctions on Sierpiński type triangles
• Isomorphism classification of infinite Sierpinski carpet graphs
• Galois coverings of Schreier graphs of groups generated by bounded automata