I once heard Lenny Susskind relate the question: "how many particles do you need in a box for the ideal gas law to 'pretty much' hold?" Obviously this question requires a notion of 'pretty much' to formalize. In thermodynamics we usually assume $N \sim N_A$, which is some astronomically large number. But it seems reasonable that if $N = 1000$, particles in a box behave more or less like a gas.
The same question can be asked of the large $N$ expansion in gauge theory. Technically, calculational tools like spectral curves only appear in the formal $N \to \infty$ limit. Presumably we understand the leading $1/N$ corrections to these predictions. My question is: how small can $N$ be such that for reasonable questions, the large $N$ limit gives more or less the correct answer?
When Lenny asked the question, it was in the context of black holes. It was essentially the question "how many degrees of freedom does a black hole need to have at a minimum for Hawking radiation to appear more-or-less thermalized?" It struck me as a very difficult question to answer, and made even more interesting by the fact that "large enough" $N$ does not seem to be that large at all.
With all that said, how badly does large $N$ gauge theory fail for groups like, say $SU(5)$? Is it completely worthless? Is it more or less accurate? How are $1/N$ corrections handled in the literature?
Aside: in the context of thermodynamics, the important "large $N$" approximation made is the Stirling approximation $\log(N!) \approx N\log(N)-N + \mathcal{O}(\log(N))$. As you can convince yourself by plotting each side of this approximation, it becomes more or less exact around $N = 4$. Obviously there is a lot more to be said than this, but I think it shows that conventional ideas about how large $N$ must be tend to be overestimates.
