Diffusion conjecture for the almost Mathieu–Markovian equation

Let $g\in\mathbb{R}$, let $\theta,\alpha\in[0,1]$, and consider the almost Mathieu–Markovian equation on $\ell^2(\mathbb{Z})$ with potential $2g\cos 2\pi(\theta+x\alpha)$ and disorder coupling $\lambda$.

Almost Mathieu diffusion conjecture. For almost every $\theta,\alpha\in[0,1]$, the equation has a diffusion constant $D(g,\lambda)\in(0,\infty)$ that is smooth for all $(g,\lambda)\in\mathbb{R}\times\mathbb{R}^+$. Moreover, $D(g,\lambda)\sim O(\lambda^2)$ for all $|g|>1$ and $D(g,\lambda)\sim O(\lambda^{-2})$ for all $|g|<1$.

This conjecture applies the expected relation between diffusion and the transport regime of the unperturbed almost Mathieu operator: the localized regime $|g|>1$ and the ballistic regime $|g|<1$ are predicted to produce opposite small-coupling scalings. The critical case $|g|=1$ is not covered by the statement.