This is quite the rabbit hole but according to some background it links to a number of interesting problems, for example phase transitions in spin-glasses in physics. The wiki page on spin glasses is a good place to start, perhaps. Here's more on background:
http://assets.press.princeton.edu/chapters/i9917.pdf
"These are all examples of what are called combinatorial optimization problems, which typically, though not always, arise from a branch of mathematics called graph theory...What have spin glasses to do with all this? As it turns out,
quite a lot. Investigations into spin glasses have turned up a number of surprising features, one of which is that the problem of finding low-energy states of spin glasses is just another one of these kinds of problems. This led directly from studies of spin glasses to the creation of new algorithms for solving the TSP [Traveling Salesman] and other combinatorial optimization problems."
The first reference in the original Kahn-Kalai “expectation threshold” conjecture (as linked in the article) is this 2005 Nature paper, "Rigorous location of phase transitions in hard optimization problems"
https://www.cs.cornell.edu/selman/papers/pdf/05.nature.phase...
> "Constraint satisfaction problems are at the heart of statistical physics,
information theory and computer science. Typically, they involve a large set of variables, each taking values in a small domain, such as {0, 1}, and a collection of constraints, each binding a few of the variables by forbidding some of their possible joint values. Examples include spin-glasses in statistical physics, error-correcting codes in information theory, and satisfiability and graph colouring in computer science. Given a collection of constraints, a fundamental scientific question is how many of them can be satisfied simultaneously."
So... the article notes "each property has what’s called a threshold: a probability at which the structure emerges, often very abruptly."
Here's a short video of a sudden phase transition in supercooled water, which is sort of comparable to a phase transition in a spin glass, or say, the transition from amorphous to crystalline silicon. These things happen at sharp thresholds but have defied first-principles calculation as far as I know about this (which isn't so much, but seems to agree with your latter question re importance):
https://www.youtube.com/watch?v=PM9nwYF1uR4
Perhaps then as a result of this work, your theoretical condensed matter physicists now have new proven mathematical tools to understand things like this?