Counting lattice animals in high dimensions

Abstract

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 ≤ d ≤ 10. From the data we compute formulas for perimeter polynomials for lattice animals of size n ≤ 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 ≤ 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.

Proper n-Cell Polycubes in n − 3 Dimensions

“…Recently, Luther and Mertens [8] provided an argument supporting this hypothesis. They showed that the highest-degree term in DX(n, n − k) is of the order 2 n n n+k−4 .…”

“…where P k (n) is a monic polynomial in n. It has also been conjectured [2,8] that the degree of P k (n) is 3k − 4. One can attempt to prove this pattern for the general case of k without actually writing down all the expressions (respective of the cases in our method), but rather by showing that one of them (containing the dominant term, that is, having the highest degree of n) has the form 2 n n n−2k−1 (n − k) times a polynomial in n of degree 3k − 4, and that all the other expressions have the same form multiplied by polynomials of at most the same degree.…”

“…(2.15)] predicted that the diagonal formula DX(n, n−k) has the pattern 2 n−2k+1 n n−2k−1 g k (n), where g k (n) is a polynomial in n. In fact, k has to be a root of g k (n) since DX(n, 0) = 0 for n > 1. Therefore, the expected form is 2 n−2k+1 n n−2k−1 (n − k)h k (n), where h k (n) is a polynomial in n. Luther and Mertens [8] provide the explicit polynomials h k (n) for 1 ≤ k ≤ 7. Careful inspection of these polynomials reveals that the leading coefficient of h k (n) has the form 2 k−1 /(k − 1)!.…”

“…In the literature of statistical physics [8,11], formulae for DX(n, n − k) are interpolated for 1 ≤ k ≤ 7. (See also a discussion by Barequet et al [2, p. 265] about how to derive these formulae from more general formulae provided by Peard and Gaunt [11].)…”

Polycubes with Small Perimeter Defect

“…Aleksandrowicz and Barequet [3] provided counts up to n = 18. Luther and Mertens [9] set the current record by computing A 3 (19), improving and extending the original algorithm of Redelmeier [10] for counting polyominoes in the plane. To-date, no formula is known for A 3 (n) (or for the number of polycubes in any dimension).…”