Counting dimer coverings on self-similar Schreier graphs

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 Sierpinski gasket.

[1]  P. W. Kasteleyn.  The Statistics of Dimers on a Lattice , 1961 .

[2]  Mikhael Gromov.  Structures métriques pour les variétés riemanniennes , 1981 .

[3]  Stephan G. Wagner,et al.  Exact and asymptotic enumeration of perfect matchings in self-similar graphs , 2009, Discret. Math..

[4]  R. Grigorchuk,et al.  From fractal groups to fractal sets , 2002, math/0202001.

[5]  Zoran Sunic,et al.  Groups St Andrews 2005: Self-similarity and branching in group theory , 2007 .

[6]  Laurent Bartholdi,et al.  On the Spectrum of Hecke Type Operators related to some Fractal Groups , 1999 .

[7]  T. Nagnibeda,et al.  Schreier graphs of the Basilica group , 2010 .

[8]  Daniele D'Angeli,et al.  Partition Functions of the Ising Model on Some Self-similar Schreier Graphs , 2010, 1003.0611.

[9]  T. Nagnibeda,et al.  Schreier graphs of the Basilica group , 2009, 0911.2915.

[10]  Shu-Chiuan Chang,et al.  Dimer Coverings on the Sierpinski Gasket , 2007, 0711.0573.

[11]  M. Fisher,et al.  Dimer problem in statistical mechanics-an exact result , 1961 .

[12]  I. Bondarenko.  Groups generated by bounded automata and their schreier graphs , 2007 .

[13]  Richard Kenyon,et al.  Lectures on Dimers , 2009, 0910.3129.

[14]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .

[15]  Stephan G. Wagner,et al.  Enumeration of matchings in families of self-similar graphs , 2010, Discret. Appl. Math..

[16]  M. Fisher.  On the Dimer Solution of Planar Ising Models , 1966 .

[17]  R. Grigorchuk.  Solved and Unsolved Problems Around One Group , 2005 .

[18]  Volodymyr Nekrashevych,et al.  Self-Similar Groups , 2005, 2304.11232.

[19]  Rostislav I. Grigorchuk,et al.  On a Torsion-Free Weakly Branch Group Defined by a Three State Automaton , 2002, Int. J. Algebra Comput..

[20]  R. Kenyon,et al.  Dimers and amoebae , 2003, math-ph/0311005.