Hugo Duminil-Copin, Stanislav Smirnov 

[The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$](http://annals.math.princeton.edu/2012/175-3/p14)

> 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 nonrigorously 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 $\text{SLE}(8/3)$.

Let $f(z) = u(x,y) + iv(x,y)$ be a holomorphic function.  The derivative should be the same regardless of how we should take the derivative.  
$$ \lim_{\Delta x \to 0} \frac{f(z + \Delta x)-f(z)}{\Delta x}
= \lim_{\Delta x \to 0} \frac{f(z + \Delta y)-f(z)}{i\Delta y} $$
Reading off the real and imaginary parts you obtain two different derivatives:
$$ \frac{\partial u}{\partial x} + i\, \frac{\partial v}{\partial x}  = -i\,\Big(\frac{\partial u}{\partial y}+ i\,\frac{\partial v}{\partial y} \big)$$
We have put a **complex structure** on the flat plane $\mathbb{C}$.  Why stop there?  Let's set:
$ dz = e^{i\theta} dx$ then:
$$ \frac{df}{dz} = e^{-i\theta} \lim_{\Delta x \to 0}\frac{f(z + e^{i\theta}\Delta x) - f(z)}{\Delta x}
= e^{-i\theta} \bigg( \cos \theta \frac{\partial }{\partial x} +  \sin \theta\; i\,\frac{\partial }{\partial y} \bigg) \bigg( u + i\, v\bigg)$$
If you let $\theta = 0$ or $\theta = \pi/2$ or $\theta = \pi/3$ this is:
$$ \bigg( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\bigg)  = \bigg(
-\frac{\partial v}{\partial y} + i \,\frac{\partial u}{\partial y}\bigg)
= e^{-i\pi/3}\bigg( (\frac{1}{2}\frac{\partial u}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial y}) + i 
(\frac{1}{2}\frac{\partial u}{\partial y} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial x})  \bigg)
 $$
The algebra looks almost right.

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Over a square lattice, all you have are approximations.  How do you know that the discrete derivative tends to the continuous complex derivative when $\Delta z \to 0$ ?  This might help us understand the limiting behavior of:

* Brownian motion as limit of random walk
* the limiting behavior of self-avoiding random walk
* loop-erased random walk (which I always mistake for self-avoiding)

[![enter image description here][1]][1]

The variable that Stanislav Smirnov and Hugo Duminil-Coupin define are not holomorphic, but parafermionic, they only obey *half* the Cauchy-Riemann equations.  

The geometric significance is still a little bit open, I think.  Overall geometric approaches to the Cauchy-Riemann equations are a bit under-valued in our modern time.


  [1]: https://i.sstatic.net/B8Lg3.png