Coordination Sequences and Critical Points

The locations of critical points of lattice models (such as Ising models, percolation problems or self-avoiding walks) depend on the topological structure of the underlying graph, in particular on the dimension and on the (mean) coordination number. However, this can only be a first approximation, and more detailed information on the graph is required for a better understanding of the precise location of critical points. For this purpose, quasiperiodic graphs prove helpful because they provide different cases with the same mean coordination number and also a check of the “universality” of this approach. Below, we display first results on the combinatorial issue to calculate the coordination sequences, and on general tendencies visible. To this end, we summarise the results for root graphs 6,7 and present the averaged coordination numbers for the rhombic Penrose and for the Ammann-Beenker tiling.