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Large $N$ limits are present in many different contexts: matrix models, gauge theories in various dimensions, conformal field theories (where $N$ is essentially the central charge).

We often hear things like "we assume that the large $N$ limit exists" (assuming maybe some scaling of the coupling constant for instance).

What does this mean precisely?

Do we mean that the large $N$ limit of any correlation function is finite?

If yes, should these correlation functions satisfy the axioms of QFT, or could they violate some of them (even if at finite $N$ there's no such violation)?

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