18 research outputs found
Automatic Proofs for Formulae Enumerating Proper Polycubes
This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions
Counting Lattice Animals in High Dimensions
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 -dimensional
hypercubic lattices for . From the data we compute formulas
for perimeter polynomials for lattice animals of size in arbitrary
dimension . When amended by combinatorial arguments, the new data suffices
to yield explicit formulas for the number of lattice animals of size
and arbitrary . 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.Comment: 18 pages, 7 figures, 6 tables; journal versio
High-dimensional holeyominoes
What is the maximum number of holes enclosed by a -dimensional polyomino
built of tiles? Represent this number by . Recent results show that
converges to . We prove that for all we have
as goes to infinity. We also construct polyominoes
in -dimensional tori with the maximal possible number of holes per tile. In
our proofs, we use metaphors from error-correcting codes and dynamical systems.Comment: 10 pages, 4 figure
Expansion in high dimension for the growth constants of lattice trees and lattice animals
We compute the first three terms of the 1/d expansions for the growth
constants and one-point functions of nearest-neighbour lattice trees and
lattice (bond) animals on the integer lattice Zd, with rigorous error
estimates. The proof uses the lace expansion, together with a new expansion for
the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the
expansion, making the present paper more self-contained with very little
change to the structure of the original paper. Accepted for publication in
Combinatorics Probability and Computin
Analyticity results in Bernoulli percolation
We prove (rigorously) that in 2-dimensional Bernoulli percolation, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that p bond c < 1/2 for certain families of triangulations for which Benjamini & Schramm conjectured that p site c ≤ 1/2