Found in 2 comments on Hacker News
PaulHoule · 2022-11-12
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Generically in the theory of phase transitions and critical phenomena the infinite dimensional case is easy because mean field theory works, but there is some dimension at which that breaks, frequently N=6, and you can write an asymptotic expansion in N-ε for the critical exponents which converges enough for most things that you can get the right answers in 3 dimensions.
We worked through this book when I was in grad school...
https://www.amazon.com/Theory-Critical-Phenomena-Introductio...
Read it in grad school, shows a lot of great uses for Feynman diagrams and such outside of elementary particle physics. Superstring theory uses 12 dimensional physics but to understand exactly how water boils you need fractional dimension physics!