Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices

Abstract

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $\lambda_S^{L^2}.$ We have made more precise the estimate of $\lambda_S,$ based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that $\lambda_S = 1.7445498 \pm 0.0000012.$ We have also studied the subdominant behaviour, and conjecture that $$ C_L(1) \sim \lambda_S^{L^2+bL+c}\cdot L^g,$$ where $b=-0.04354 \pm 0.0001,$ $c=0.5624 \pm 0.0005,$ and $g=0.000 \pm 0.005.$ We implemented a very efficient algorithm for enumerating paths on the square and hexagonal lattices making use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths. Using this algorithm we extended and then analysed series for SAWs spanning the square lattice and self-avoiding polygons (SAPs) crossing the square lattice. These are known to also grow as $\lambda_S^{L^2}.$ The sub-dominant term $\lambda^b$ is found to be the same as for SAWs crossing the square, while the exponent $g = 1.75\pm 0.01$ for spanning SAWs and $g = -0.500 \pm 0.005$ for SAPs. We have also studied the analogous problems on the hexagonal lattice, and generated series for a number of geometries. In particular, we study SAWs and SAPs crossing rhomboidal, triangular and square domains on the hexagonal lattice, as well as SAWs spanning a rhombus. We estimate that the analogous growth constant $\lambda_H=1.38724951 \pm 0.00000005,$ so an even more precise estimate than found for the square lattice. We also give estimates of the sub-dominant terms.