A numerical adaptation of self-avoiding walk identities from the honeycomb to other 2D lattices

Abstract

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis that the connective constant of self-avoiding walks (SAWs) on the honeycomb lattice is A key identity used in that proof depends on the existence of a parafermionic observable for SAWs on the honeycomb lattice. Despite the absence of a corresponding observable for SAWs on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed on the honeycomb lattice persist on other lattices. This permits the accurate estimation, though not an exact evaluation, of certain critical amplitudes, as well as critical points, for these lattices. For the honeycomb lattice, an exact amplitude for loops is proved.