Random Surfaces Hide an Intricate Order

It's usually a fair bet that if a writer on the general internet uses ambiguous units, date formatting, or seasons, it's the American ones.

Agree. Aside from coefficients/ratios, a number without units is meaningless.

How do you know it's not a typically used unit? It could be Celsius at high pressure.

Or maybe you, like almost everyone else, aren't confused at all because you know water at sea level boils at 212 degrees Fahrenheit, a unit which is typically used in the U.S., and it's an informal quote by a U.S. resident of a coastal city.

That's both a quote, and an analogy. You're not wrong, but it's hardly relevant.

It's not an intricate pattern/geometric order. It's a transition in the "connectedness" of the structure.

I think it means order in a way that's quite specific to the study of phase transitions. In a phase transition a system switches between a disordered phase (eg a gas) and an ordered phase (eg a solid) as measured by a so-called 'order parameter'. Here the transition is that of percolation (google it, or percolation theory, for detail - it's a big subject by itself) which, in 2 dimensions say, transitions between a phase where the order parameter is zero and there are just disconnected clusters with an expontential size distribution, and a phase where the order parameter is non-zero and a cluster is connected across the entire system. The critical point at which the transition actually occurs tends to be the point of most interest and it's characterised by power law distributions. So it's a slightly broader definition of order than people outside the subject might be accustomed to.

Here, they analyse percolation on a random geometry and look at how this influences the percolation transition. For instance, how the size of the largest cluster scales with system size. This isn't new in itself, it's been an interesting problem for at least a couple of decades. Just skimming the paper, I _think_ what's new here is one or two new results for the combination of this particular random geometry and percolation. I have to say, I didn't quite get the sense of novelty from the paper that I felt the title of the article promised.