Question

Lattice and bridge network

Answer 1

It's solved as physics since the late 1970s, but in mathematics it is impossible to formalize the proof. You can define the lattice gauge theory, and it is a statistical system in Euclidean space defined by Wison (also by Polyakov), and it is clearly gapped because it randomizes in simulations at large distanes, and the strong coupling expansion shows it stays random, but to prove it has a continuum limit requires linking the heuristics for short distances, which is asymptotic freedom, to the heuristics for long distances, which are the the gauge field randomizes, and this requires a method of defining statistical field systems and defining the RG flow.

I will assume that you work in a universe where measure is universal, so I can talk about statistics without Borel sets and nonsense like this. On any lattice, you can defined the gauge theory. Then you take the limit as the lattice size gets small and the coupling gets weak is that you should take this limit keeping the randomization scale fixed.

That this is possible requires a rigorous construction of the renormalization flow which keeps the statistical fluctuations at long distances fixed. This is not so hard to do, even formally. But the difficulty is in proving that the flow is one dimensional no matter how you make a lattice approximation, and no matter how you take the limit, that it matches onto the perturbative calculation at short distances, and it matches onto the strong coupling expansion at long distances. It is infuriating, because it is manifestly obvious if you ever simulated lattice QCD.