I heard many times phrases like "large $N$ is a clever way to organize the diagrammatic expansion" or "each diagram in the large $N$ expansion contains an infinite number of usual perturbative diagrams".
I also know that for each order in $N$, the series in the 't Hooft coupling parameter $t$ has better convergence properties that the perturbative expansion in the usual coupling constant - due to the smaller number of diagrams (their number is growing polynomially, not factorially).
Can somebody explain the relation between these two approaches? For example, it would be great to see which perturbative diagrams are contained in some leading-order $1/N$ diagram, and how/why they are resummed.
Any references appreciated as well.
UPDATE
OK, let me try to formulate what I understood by now, please tell me if smth is wrong. So, assume the Lagrangian looks smth like this: \begin{equation} \mathcal{L} = \dfrac{1}{g} \left( (\partial \Phi)^2 + \Phi^4 \right) = \dfrac{t}{N} \left( (\partial \Phi)^2 + \Phi^4 \right) \end{equation}
Where \begin{equation} g N = t \end{equation}
The expansions for, say, partition function in terms of $g$ and $\{t, 1/N\}$ have form: \begin{equation}\begin{alignedat}{9} \mathcal{Z} &= &&\alpha_0 + \alpha_1 g + \alpha_2 g^2 + \ldots \\ &= N &&(a_{0} + a_{1} t + a_{2} t^2 +\ldots)\\ &+ &&(b_{0} + b_{1} t + b_{2} t^2 +\ldots)\\ &+ \dfrac{1}{N}&&(c_{0} + c_{1} t + c_{2} t^2 +\ldots) + \ldots\\ &= N &&(a_{0} + a_{1} (g N) + a_{2} (g N)^2 +\ldots)\\ &+ &&(b_{0} + b_{1} (g N) + b_{2} (g N)^2 +\ldots)\\ &+ \dfrac{1}{N}&&(c_{0} + c_{1} (g N) + c_{2} (g N)^2 +\ldots) + \ldots\\ \end{alignedat}\end{equation}
Which gives us for the coeficients of the $g$-series: \begin{equation}\begin{alignedat}{9} \alpha_0 &= N &&a_0 + b_0 + \dfrac{1}{N} c_0 + \dfrac{1}{N^2} d_0 + \ldots \\ \alpha_1 &= N &&\left( a_1 + b_1 + \dfrac{1}{N} c_1 + \dfrac{1}{N^2} d_1 + \ldots \right) \\ \alpha_2 &= N^2 &&\left( a_2 + b_2 + \dfrac{1}{N} c_2 + \dfrac{1}{N^2} d_2 + \ldots \right) \\ \end{alignedat}\end{equation}
My understanding is that for some reason at large power of $g$ and $t$ the coefficients behave as follows (I actually have some understanding about the factorial growth, but have not learned about the $1/N$ case yet). \begin{equation}\begin{alignedat}{9} \alpha_k &\propto k! \\ a_k,b_k,c_k &\propto \text{(some power of $k$)} \end{alignedat}\end{equation}
I'd like to know if the last statement is correct, and if it's possible to see it from the previous line.