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I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are only available for some classical groups, namely the orthogonal, unitary and symplectic groups. However, my understanding of the derivation of these formulae leads me to conjecture that the asymptotics of the moments for any compact Lie group $\mathcal{G}$ are inversely polynomial in the dimension of the largest irreducible sub-representation of $\mathcal{G}$. I have however been unable to find any results to this effect.

Does any one know of such results, or any counterexamples to the conjecture above? I am interested only in the asymptotic dependence on the dimension parameters and not in the explicit form of the integrals.

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The generalization of Weingarten calculus to compact Lie groups is studied in Expectation values of polynomials and moments on general compact Lie groups, see section 4.

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