Critical phenomena on self-similar lattices

Recently, methods for computing the number of embeddings of graphs in self-similar structures were proposed, allowing the use of the series-expansion technique for statistical systems on these geometries. In comparison to other techniques, the series-expansion method provides the most reliable results whose accuracy can be improved in a systemic way by increasing the order of the series. The graph counting method can be used in a fairly general family of self-similar deterministic fractals which are obtained by the iteration of a fixed rule of construction. It enables one to obtain the behavior of statistical system on fractal lattices which have the same fractal dimension but different lacunarities, allowing a complete study of those system on non-Euclidean lattices. In this paper, we apply the graph counting method to find the coefficients of the high-temperature series expansions for the susceptibility of the Ising model on Sierpinski carpets.

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