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

@article{DuminilCopin2010TheCC,
title={The connective constant of the honeycomb lattice equals \$\sqrt\{2+\sqrt2\}\$},
author={Hugo Duminil-Copin and Stanislav Smirnov},
journal={arXiv: Mathematical Physics},
year={2010}
}
• Published 4 July 2010
• Mathematics
• arXiv: Mathematical Physics
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…
140 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

### Discretely holomorphic parafermions and integrable loop models

• Mathematics
• 2008
We define parafermionic observables in various lattice loop models, including examples where no Kramers–Wannier duality holds. For a particular rhombic embedding of the lattice in the plane and a

### Conformal invariance of planar loop-erased random walks and uniform spanning trees

• Mathematics
• 2001
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $$D\mathop \subset \limits_ \ne \mathbb{C}$$ is equal to the radial SLE2 path. In particular, the

### Universality in the 2D Ising model and conformal invariance of fermionic observables

• Mathematics, Physics
• 2009
It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no

### Towards conformal invariance of 2D lattice models

Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but

### Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the

### The self-avoiding walk

• Physics
• 1991
The self-avoiding walk is a mathematical model with important applications in statistical mechanics and polymer science. This text provides a unified account of the rigorous results for the

### On the scaling limit of planar self-avoiding walk

• Mathematics
• 2002
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In

### Exact Critical Point and Critical Exponents of O ( n ) Models in Two Dimensions

A two-dimensional $n$-component spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model. Subject to some plausible assumptions this leads to an analytic calculation of the

### Principles of Polymer Chemistry

• A. Ravve
• Materials Science, Chemistry
• 1995
Physical Properties and Physical Chemistry of Polymers.- Free-Radical Chain-Growth Polymerization.- Ionic Chain-Growth Polymerization.- Ring-Opening Polymerizations.- Common Chain-Growth Polymers.-