Abstract
A fractal latticeF is defined here to comprise all points of the forma + ma′+ m2 a″+ ... +mqa(q), whereq is a nonnegative integer anda, a′,..., a(q)∈A, whereA is a finite set of points in some Euclidean space. Providedm is not too small (in particular,m must be at least 2), the dimension ofF is shown to beD = log n/logm, wheren is the number of points inA. It is shown further that an Ising model onF, with a ferromagnetic pair interaction r−α between spins separated by a distancer, has a phase transition ifD < α < 2D. On the other hand, for α > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.
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Penrose, O. Phase transitions on fractal lattices with long-range interactions. J Stat Phys 45, 69–88 (1986). https://doi.org/10.1007/BF01033078
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DOI: https://doi.org/10.1007/BF01033078