Counting dimer coverings on self-similar Schreier graphs

TL;DR

This paper investigates the enumeration of dimer coverings on self-similar graphs derived from group actions on trees, focusing on their partition functions and connections to fractals like the Sierpiński gasket.

Contribution

It introduces a framework for analyzing dimer models on self-similar graphs generated by group actions, with detailed study of the Hanoi Towers group example.

Findings

Partition functions for dimer models on self-similar graphs are characterized.

Connections between graph weights and group automorphisms are established.

Results relate to fractal structures such as the Sierpiński gasket.

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 Sierpi\'nski gasket.