Statistics of close-packed dimers on fractal lattices

TL;DR

This paper analyzes the enumeration and entropy of close-packed dimers on a family of fractal lattices, revealing how entropy varies with lattice parameters and comparing different fractal structures.

Contribution

It provides exact recurrence enumeration results and entropy calculations for dimers on modified rectangular and 4-simplex fractal lattices, expanding understanding of dimer statistics on fractals.

Findings

Entropy per dimer increases with the scaling parameter p.

Entropy per dimer is lower on MR fractals than on 4-simplex lattices.

Results are compared with previous studies on translationally invariant and other fractal lattices.

Abstract

We study the model of close-packed dimers on planar lattices belonging to the family of modified rectangular (MR) fractals, whose members are enumerated by an integer $p\geq 2$, as well as on the non-planar 4-simplex fractal lattice. By applying an exact recurrence enumeration method, we determine the asymptotic forms for numbers of dimer coverings, and numerically calculate entropies per dimer in the thermodynamic limit, for a sequence of MR lattices with $2\leq p\leq8$ and for 4-simplex fractal. We find that the entropy per dimer on MR fractals is increasing function of the scaling parameter $p$, and for every considered $p$ it is smaller than the entropy per dimer of the same model on $4$-simplex lattice. Obtained results are discussed and compared with the results obtained previously on some translationally invariant and fractal lattices.