Exact Solution of a Directed-Site Animals-Enumeration Problem in Three Dimensions

Abstract

An exact equivalence is established between the directed-site animals-enumeration (DSAE) problem in $d$ dimensions and a crystal-growth model defined on the same lattice. In special cases, the latter reduces to the calculation of the free energy of a ($d\ensuremath{-}1$)-dimensional lattice gas with extended hard cores. The author solves a DSAE problem with $d=3$ exactly by using its equivalence to the hard-hexagon problem, and shows that the exponent $\ensuremath{\theta}=\frac{5}{6}$. A new and simpler solution to the DSAE problem on the square and triangular lattices is given.