Noise stability

Definition

For [math]f:\{-1,1\}^{n}\rightarrow\mathbb{\mathbb{R}} [/math] and [math]\rho∈[−1,1][/math], the noise stability of [math]f[/math] at [math]\rho[/math] is [math]Stab_{\rho}[f]=\underset{\underset{\rho-correlated}{(x,y)}}{\mathbb{E}}[f(x)f(y)][/math], where [math] x [/math] and [math]y [/math] are [math]\rho[/math]-correlated if [math] y_{i}=\begin{cases} x_{i} & with\space prabability\space \frac{1}{2}+\frac{1}{2}\rho\\ -x_{i} & with\space prabability\space \frac{1}{2}-\frac{1}{2}\rho \end{cases}[/math]