Let $V_{\lambda}$ and $W_{\lambda}$ be the irreducible representations of $S(n)$ and $\mathfrak{su}(N,\mathbb{C})$ associated to the partition $\lambda \in \mathbb{Y}$ of size $| \lambda |=n$ and length $l(\lambda) \leq N$. The following limit $$\frac{\dim V_{\lambda}}{n!} = \lim_{N \rightarrow \infty} \frac{\dim W_{\lambda}}{N^{n}}$$ follows immediately from the well known hook (content) formulas $$\dim V_{\lambda} = \prod_{\square \in \lambda} \frac{n!}{h(\square)} \ \ \ \ \dim W_{\lambda} = \prod_{\square \in \lambda} \frac{N + c(\square)}{h(\square)}$$ which can be found in Macdonald's book. Notice that $n! = \dim_{\mathbb{C}} \mathbb{C}[S(n)]$ and $N^n = \dim_{\mathbb{C}} (\mathbb{C}^N)^{\otimes n}$, so what we're seeing is that as $N \rightarrow \infty$, the relative multiplicity of $V_{\lambda}$ in Schur-Weyl duality approaches the relative multiplicity of $V_{\lambda}$ in the regular representation.

Does anyone have a good feeling for why this is true?

Also, let us not forget the Peter-Weyl theorem! If for a compact group $G$ we write $G^{\vee}$ for its set of finite dimensional irreducible representations over $\mathbb{C}$, we have $$L^2(SU(N)) = \widehat{\bigoplus_{\lambda \in SU(N)^{\vee}}} W_{\lambda} \boxtimes W_{\lambda} $$ $$(\mathbb{C}^N)^{\otimes n}=\bigoplus_{\lambda \in SU(N)^{\vee} \cap S(n)^{\vee}} V_{\lambda} \boxtimes W_{\lambda} $$ $$\mathbb{C}[S(n)] = \bigoplus_{\lambda \in S(n)^{\vee}} V_{\lambda} \boxtimes V_{\lambda}$$

The limit we discussed above relating the second to the third line here actually also happens when we pass from the first to the second line: the ``relative multiplicity'' of $W_{\lambda}$ in its regular representation approaches the relative multiplicity of $W_{\lambda}$ in Schur-Weyl duality.

Can anyone give me some intuition for what's going on here + why I might expect such a result?