Coloring decay

Recursion

[math]\displaystyle{ q+1 }[/math] color, [math]\displaystyle{ d }[/math]-degree

• recursion:

[math]\displaystyle{ f(x,y)=\frac{qy^d}{x^d+qy^d} }[/math]

• fixed point:

[math]\displaystyle{ \hat{x}=\hat{y}=\frac{q}{q+1} }[/math]

• partials:

[math]\displaystyle{ \begin{align} \frac{\partial f(x,y)}{\partial x}&=-\frac{dqx^{d-1}y^d}{(x^d+qy^d)^2}\\ \frac{\partial f(x,y)}{\partial y}&=\frac{dqx^{d}y^{d-1}}{(x^d+qy^d)^2} \end{align} }[/math]

at the fixed point [math]\displaystyle{ \frac{\partial f(\hat{x},\hat{y})}{\partial x}=-\frac{d}{q+1} }[/math] and [math]\displaystyle{ \frac{\partial f(\hat{x},\hat{y})}{\partial y}=\frac{d}{q+1} }[/math].

• ratio:

[math]\displaystyle{ \alpha(x,y)=\left[-\frac{\partial f(x,y)}{\partial x}\frac{1}{\Phi(x)}+\frac{\partial f(x,y)}{\partial y}\frac{1}{\Phi(y)}\right]\Phi\left(f(x,y)\right) }[/math]

• uniqueness:

at the fixed point [math]\displaystyle{ \hat{x}=\hat{y}=\frac{q}{q+1} }[/math], [math]\displaystyle{ \alpha(\hat{x},\hat{y})=\frac{2d}{q+1} }[/math]. The critical boundary of uniqueness is [math]\displaystyle{ 2d=q+1 }[/math].

Cancelation

Let the system be right at the critical boundary, i.e. [math]\displaystyle{ 2d=q+1 }[/math]. Then [math]\displaystyle{ \alpha(\hat{x},\hat{y})=1 }[/math].

[math]\displaystyle{ \begin{align} \left.\frac{\partial\alpha(x,y)}{\partial x}\right|_{x=y=\frac{q}{q+1}} &= \left.\left[-\frac{\partial^2 f(x,y)}{\partial x^2}\frac{1}{\Phi(x)}+\frac{\partial f(x,y)}{\partial x}\frac{\Phi'(x)}{(\Phi(x))^2}+\frac{\partial^2 f(x,y)}{\partial y\partial x}\frac{1}{\Phi(y)}\right]\Phi(f(x,y))\right|_{x=y=\frac{q}{q+1}}\\ &\quad\,\,+\left.\left[-\frac{\partial f(x,y)}{\partial x}\frac{1}{\Phi(x)}+\frac{\partial f(x,y)}{\partial y}\frac{1}{\Phi(y)}\right]\Phi'\left(f(x,y)\right)\frac{\partial f(x,y)}{\partial x}\right|_{x=y=\frac{q}{q+1}}\\ &= -\frac{\partial^2 f(\hat{x},\hat{y})}{\partial x^2}+\frac{\partial^2 f(\hat{x},\hat{y})}{\partial y\partial x}-\frac{\Phi'(\hat{x})}{\Phi(\hat{x})}. \end{align} }[/math]

However,

[math]\displaystyle{ \begin{align} \left.\frac{\partial\alpha(x,y)}{\partial y}\right|_{x=y=\frac{q}{q+1}} &= \left.\left[-\frac{\partial^2 f(x,y)}{\partial x\partial y}\frac{1}{\Phi(x)}+\frac{\partial^2 f(x,y)}{\partial y^2}\frac{1}{\Phi(y)}-\frac{\partial f(x,y)}{\partial y}\frac{\Phi'(y)}{(\Phi(y))^2}\right]\Phi(f(x,y))\right|_{x=y=\frac{q}{q+1}}\\ &\quad\,\,+\left.\left[-\frac{\partial f(x,y)}{\partial x}\frac{1}{\Phi(x)}+\frac{\partial f(x,y)}{\partial y}\frac{1}{\Phi(y)}\right]\Phi'\left(f(x,y)\right)\frac{\partial f(x,y)}{\partial y}\right|_{x=y=\frac{q}{q+1}}\\ &= \left[-\frac{\partial^2 f(\hat{x},\hat{y})}{\partial x\partial y}\frac{1}{\Phi(\hat{x})}+\frac{\partial^2 f(\hat{x},\hat{y})}{\partial y^2}\frac{1}{\Phi(\hat{x})}-\frac{1}{2}\frac{\Phi'(\hat{x})}{(\Phi(\hat{x}))^2}\right]\Phi(\hat{x}) +\left[\frac{1}{2}\frac{1}{\Phi(\hat{x})}+\frac{1}{2}\frac{1}{\Phi(\hat{x})}\right]\frac{\Phi'(\hat{x})}{2}\\ &= -\frac{\partial^2 f(\hat{x},\hat{y})}{\partial x\partial y}+\frac{\partial^2 f(\hat{x},\hat{y})}{\partial y^2}. \end{align} }[/math]