Several papers such as this (warning, PDF) and this (PDF again) talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (which, if you read those papers, gives rise to some neat effects).
I'm having trouble understanding how, though, aside from just the math. The 2nd paper mentions a more qualitative interpretation:
Near the percolation threshold the metallic clusters are separated by thin dielectric regions. Each pair of nearest clusters forms a condenser [i.e., capacitor] whose effective surface tends to infinity near the percolation threshold. Then the effective capacity of the system diverges, too.
Ok, let's assume for simplicity the "dielectric" here is vacuum, and use the form of a parallel plate capacitor, $C=A\epsilon_0/d$. By "effective surface tends to infinity", I'm guessing they mean that $d$ is going to zero (because the distance between clusters is going to 0), and thus $C$ is increasing to infinity.
From there, I'm a little confused. My best guess is that the impedance goes as $Z\propto 1/(i\omega C)$, and conductivity $\sigma \propto 1/Z$, and dielectric constant $\epsilon \propto i\omega \sigma$, so when the C increases, Z decreases, the imaginary part of $\sigma$ increases, and thus the real part of $\epsilon$ increases.
But I'm not really sure about that, and I'm having trouble reconciling it with what must happen after the percolation transition.
Does anyone have a better physical understanding of this?