Rooted Spiral Trees on Hyper-cubical Lattices

We study rooted spiral trees in 2, 3 and 4 dimensions on a hyper cubical lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant λ. Series expansions give θ = −1.3667 ± 0.001 and ν = 1.3148 ± 0.001 on a square lattice. With Monte-Carlo simulations we get the estimates as θ = −1.364 ± 0.01, and ν = 1.312 ± 0.01. These results are numerical evidence against earlier proposed dimensional reduction by four in this problem. In dimensions higher than two, the spiral constraint can be implemented in two ways. In either case, our series expansion results do not support the proposed dimensional reduction. Spiral structures are very common in nature. Some examples of the beautiful spiral structures in galaxies, shoot arrangement in plants, polymers with spiral structure etc may be found in the book by Hargittai [1]. In statistical mechanics, lattice models of spiral self avoiding walks have been studied and can be solved exactly in two dimension [2, 3], though no solution is known for the self avoiding walks without the spiral constraint. A model of spiral trees and animals was proposed by Li and Zhou [4], which based on numerical studies was suggested to be in a new universality class. This problem was further studied by Bose et. al [5]. Based on the numerical evidence, and guided by the fact that magnetic field acting perpendicular to the motion of a charged particle produces spiralling motion and reduction by two in effective dimensionality, they conjectured that