Combinatorics of 3D directed animals on a simple cubic lattice
Abstract
We provide combinatorial arguments based on a two-dimensional extension of a locally-free semigroup allowing us to compute the growth rate, $\Lambda$, of the partition function $Z_N=N^{\theta}\Lambda^N$ of the $N$-particle directed animals ($N\gg 1$) on a simple cubic lattice in a three-dimensional space. Establishing the bijection between the particular configuration of the lattice animal and a class of equivalences of words in the 2D projective locally-free semigroup, we find we find $\ln \Lambda = \lim_{N\to\infty} \ln Z_N / N$ with $\Lambda= 2(\sqrt{2}+1) \approx 4.8284$.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2020
- DOI:
- 10.48550/arXiv.2002.00618
- arXiv:
- arXiv:2002.00618
- Bibcode:
- 2020arXiv200200618N
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- High Energy Physics - Theory;
- Mathematics - Combinatorics
- E-Print:
- We have realized that "Mikado ordering" valid in 2D fails in 3D. We found source of error and have proposed a new approach for enumeration of 3D heaps of pieces based on a nontrivial relation to the 2D hard-core lattice gas. We would like to withdraw the paper because the replacement could be confusing: we do not make modifications of a former approach, but replace it with a principally new one