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confusion about the Cauchy-Riemann equation
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john mangual
john mangual
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Hugo Duminil-Copin, Stanislav Smirnov

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

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: $$ \bigg( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\bigg) = \bigg( -\frac{\partial v}{\partial y} + i \,\frac{\partial u}{\partial y}\bigg) = e^{-i\pi/3}\bigg( (\frac{1}{2}\frac{\partial u}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial y}) + i (\frac{1}{2}\frac{\partial u}{\partial y} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial x}) \bigg) $$$$ \bigg( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\bigg) = \bigg( -\frac{\partial v}{\partial y} + i \,\frac{\partial u}{\partial y}\bigg) = e^{-i\pi/3}\bigg(( \frac{1}{2}\frac{\partial u}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial y}) + i (\frac{1}{2}\frac{\partial v}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial u}{\partial y} )\bigg) $$ The algebra looks almost right.

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

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.

Hugo Duminil-Copin, Stanislav Smirnov

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

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.

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

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.

Hugo Duminil-Copin, Stanislav Smirnov

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

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: $$ \bigg( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\bigg) = \bigg( -\frac{\partial v}{\partial y} + i \,\frac{\partial u}{\partial y}\bigg) = e^{-i\pi/3}\bigg(( \frac{1}{2}\frac{\partial u}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial v}{\partial y}) + i (\frac{1}{2}\frac{\partial v}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial u}{\partial y} )\bigg) $$ The algebra looks almost right.

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

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.

Source Link
john mangual
john mangual
  • 22.8k
  • 4
  • 63
  • 172

Hugo Duminil-Copin, Stanislav Smirnov

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

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.

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

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.