The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

TL;DR

This paper provides the first rigorous proof that the connective constant of the hexagonal lattice is exactly 2+2, confirming a value previously derived through non-rigorous physics methods.

Contribution

The paper introduces a rigorous mathematical proof for the connective constant of the hexagonal lattice using parafermionic observables, bridging physics conjecture and mathematical validation.

Findings

Proof confirms the connective constant as 2+2 for the hexagonal lattice.

Uses parafermionic observable satisfying discrete Cauchy-Riemann relations.

Establishes a foundation for potential convergence of self-avoiding walk to SLE(8/3).

Abstract

We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).