Rooted Spiral Trees on Hyper-cubical lattices

TL;DR

This paper investigates rooted spiral trees on hyper-cubic lattices across multiple dimensions, providing exact bounds, series expansions, and Monte Carlo results that challenge previous theories of dimensional reduction.

Contribution

It offers the first comprehensive numerical analysis of rooted spiral trees in higher dimensions, refuting earlier claims of dimensional reduction by four.

Findings

Exact lower bound of 1.93565 on growth constant in 2D

Series expansions yield specific critical exponents

Monte Carlo results support series expansion conclusions

Abstract

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 $\lambda$. Series expansions give $\theta=-1.3667\pm 0.001$ and $\nu = 1.3148\pm0.001$ on a square lattice. With Monte-Carlo simulations we get the estimates as $\theta=-1.364\pm0.01$, and $\nu = 1.312\pm0.01$. 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.