# How large is large $N$?

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.

• We don't know how to solve SU(n) QCD neither for finite n, nor in the infinite limit, so we cannot say how good large-n is for n=5. We don't know what either of those theories look like, really. But, hopefully, n=3 is already "large". Mar 8, 2021 at 22:03
• To be clear, I'm not asking for a "solution" of Yang-Mills theory (lol). I'm asking how predictions made in the 't Hooft limit compare to actual perturbation theory calculations, and if there is a quantitative way to compare them. We definitely do know a good deal about SU(N) YM in the 't Hooft limit... Mar 8, 2021 at 22:18
• The Stirling formula is a good approximation for N=4 if the factor $\sqrt{2\pi N}$ is used. Only using $Nlog(N) - N$ the difference doesn't converge to zero. But of course the relative error does converge. Mar 8, 2021 at 23:45
• @hulsey You might be interested in these papers: SU(N) gauge theories in four dimensions: exploring the approach to N = infinity, and also SU(N) gauge theories in 2+1 dimensions. They report some interesting numerical results about how good the large $N$ approximation is, using a little bit of extrapolation. Witten's 1979 paper Baryons in the $1/N$ expansion is also required reading (maybe you've already read it), even though it only speculates about your question. Mar 9, 2021 at 1:56