Rooted Spiral Trees on Hyper-Cubic Lattices
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We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubic 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±0.0010 and ν = 0.6574±0.0010 on a square lattice. With Monte-Carlo simulations we get the estimates as θ= −1.364±0.010, and ν = 0.656±0.010. 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.