Percolation, i.e. the study of things similar to liquids dripped onto porous media, is an unnecessary fascinating topic. But you do not need to dive into the mathematics  to enjoy this fascination.
Morever, we find that percolation is a very useful model, and on top of that, it easy to explain as well.
The main feature is that there are two phases, the subcritical phase and the supercritical phase.
Imagine a drip coffee maker. In the beginning, the coffee powder in the filter is dry, and nothing is pouring into the can below the filter. The coffee machine starts dripping water onto the coffee grounds, but each drop seemingly vanishes inside the medium. And this takes a while. This is the subcritical phase. There are isolated wet parts in the coffee grounds, but that's it.
But at some point something happens. Then there is enough water in the filter that our drops do not simply get soaked up by the coffee grounds, but they flow through the damp coffee grounds and drip down into the can. Suddenly everything is so wet, it seems that like very drop just immediately flows through the medium. This is the supercritical phase. Now there are at best some isolated dry parts.
Criticality now is the phase transition itself. Depending on the lingo, it could also be called singularity.
Assume now for some reason, we want to prevent coffee from forming in the can. In the subcritical phase, we could simply scoop away the parts that just got dripped on. Or they just dry out, depending on the influx of water. But in the supercritical phase, even if we were to scoop away those parts, there would be enough fluid in the medium that coffee would continue to pour down the can.
Now, we have all seen this in other areas of life. Early stage pandemics follow the rules of percolation models. At first there are only a few isolated infections, and as long as you can somehow reduce the amount of water dripping into your coffee filter, you could keep it at these isolated pockets and scoop out the wet parts until it dies out. It's the easiest, safest and cheapest way to fend off such a catastrophe, but of course, requires rigorous effort on many fronts, which is hard to sustain for genpop.
We are in much better luck on our own. Our immune system also is like a giant coffee filter, and incoming pathogens usually just dry out as they get defeated by our immune cells. Only when they replicate too much for our immune system to catch up, we actually get into trouble and enter the supercritical phase and become sick.
Knowledge is in some sense another nice example, as it also shows the limits of this simple model. Usually scientific advances or more nuanced views do not really have an impact on our lives, until there is a sufficient saturation of the population. Here we probably want to evolve the model a bit from one drip, to multiple sources. While knowledge spreads only from a few points, propaganda is like hosing down the whole medium. But since actors inside a population can move around and once they are wet, they can start dripping at other points, things quickly become too complicated to be captured in a simple percolation model.
Creating non-examples to a model is somewhat hard, because either they are trivial or too artificial. This is much easier with very overused memes, where people apply models to things that do not fit the model. And percolation models are not really overhyped. Indeed, I mainly wrote this post, because I like percolation models and they are not really known, so there is also not much to play us out.
|||btw. you can pretty straightforwardly solve the PME on manifolds with conical singularities using the heat kernel mapping properties outlined in https://arxiv.org/abs/1605.03935 but then you're like done after 4 pages and wonder what to do with it|