Counting Lattice Animals in High Dimensions

TL;DR

This paper implements an efficient algorithm to enumerate lattice animals in high-dimensional lattices, extending existing data, deriving formulas, and providing numerical estimates that align with theoretical predictions.

Contribution

It introduces a fast implementation of Redelemeier's algorithm, extends enumeration tables, and derives explicit formulas for lattice animals in high dimensions.

Findings

Extended tables of lattice animal counts for 3 to 10 dimensions.

Derived explicit formulas for lattice animals of size up to 14 in arbitrary dimensions.

Numerical estimates for growth rates and exponents match Monte Carlo and field theory predictions.

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 $3 \leq d\leq 10$. From the data we compute formulas for perimeter polynomials for lattice animals of size $n\leq 11$ 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 $n\leq 14$ 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.