Counting Lattice Animals in High Dimensions

Abstract

We present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter polynomials and series expansion coefficients in $d$-dimensional hypercubic lattices for $3\le d\le 10$. From the data we compute formulas for perimeter polynomials for lattice animals of size $n\le 11$ in arbitrary dimension $d$. When amended by combinatorial arguments, the new data suffices to yield explicit formulas for the number of lattice animals of size $n\le 14$ and arbitrary $d$. We also use the enumeration data to compute numerical estimates for growth rates and exponents in high dimensions that agree very well with Monte Carlo simulations and recent predictions from field theory.

18 pages, 7 figures, 6 tables; journal version

References in corpus (3)

• Simulations of lattice animals and trees
• Exactly solved models of polyominoes and polygons
• Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers

Cited by in corpus (8)

• Yang-Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 <= d <= 7
• Renormalization group for measurement and entanglement phase transitions
• Series Expansion of the Percolation Threshold on Hypercubic Lattices
• The design of efficient algorithms for enumeration
• The Perimeter of Proper Polycubes
• Plateau Polycubes and Lateral Area
• Lattice paths inside a table, II
• Directed Plateau Polyhypercubes