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Recently, Duminil-Copin and Smirnov proved a long-standing conjecture byNienhuis that the connective constant of self-avoiding walks on the honeycomblattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends onthe existence of a parafermionic observable for self-avoiding walks on thehoneycomb lattice. Despite the absence of a corresponding observable for SAW onthe square and triangular lattices, we show that in the limit of largelattices, some of the consequences observed on the honeycomb lattice persist onother lattices. This permits the accurate estimation, though not an exactevaluation, of certain critical amplitudes, as well as critical points, forthese lattices. For the honeycomb lattice an exact amplitude for loops isproved.

A numerical adaptation of SAW identities from the honeycomb to other 2D lattices