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 $\Lambda =2(\sqrt{2}+1)\approx 4.8284$.

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