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

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