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

TL;DR

This paper extends the analysis of self-avoiding walks from the honeycomb lattice to other 2D lattices, providing numerical estimates of critical parameters despite lacking exact observables.

Contribution

It introduces a numerical adaptation of SAW identities to square and triangular lattices, enabling estimation of critical points and amplitudes without exact formulas.

Findings

Estimated critical points for square and triangular lattices.

Confirmed persistence of some honeycomb lattice properties in large lattices.

Proved an exact amplitude for loops on the honeycomb lattice.

Abstract

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by Nienhuis that the connective constant of self-avoiding walks on the honeycomb lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks on the honeycomb lattice. Despite the absence of a corresponding observable for SAW 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.