Let $t$ be a natural number. For a unitary matrix $U$ let $U_{1,1}$ be the top left matrix element of $U$. I am trying to figure out the value of $\int |U_{1,1}|^{2t} dH(U)$ where $H$ is the Haar measure over the unitary group.
I know that this integral could be expressed as $$\sum_{\sigma,\tau\in S_t} \mbox{Wg}_d(\sigma \tau^{-1}) = t!\sum_{\sigma\in S_t} \mbox{Wg}_d(\sigma)$$ where $d$ is the implicit dimension the unitary group acts on $\mbox{Wg}_d$ is the $d$-dimensional Weingarten function, but I have no idea how to calculate this sum.
What I can say is that (applying Weingarten's theorem in reverse) $$\sum_{\sigma\in S_t} \mbox{Wg}_d(\sigma) = \int U_{1,2} U^\dagger_{2,1}\ldots U_{1,t+1} U^\dagger_{t+1,1} dH(U),$$ and for $d \gg t$ I think this should become well approximated by $$\int U_{1,2} U^\dagger_{2,1} dH(U) \ldots \int U_{1,t+1} U^\dagger_{t+1,1}dH(U)$$ (intuitively, having a large dimension makes the correlation between a small subset of matrix elements weaker). The value of $\int U_{1,2} U^\dagger_{2,1} dH(U)$ is exactly the expected absolute value of some element of a random complex unit vector, which again I am not sure how to calculate.
Best Answer
We have $$\mathbb{E}_{U(N)} |U_{1,1}|^{2t} = \binom{t+N-1}{t}^{-1}$$ for all integers $t \ge 1$. A reference is Corollary 1.2 of J. Novak, ``Truncations of random unitary matrices and Young tableaux'', Electron. J. Combin. 14 (2007), no. 1, Research Paper 21. He indeed uses a Weingarten-type approach, as part of a much more general result than this specific corollary. He studied moments of $\mathrm{Tr}(U')$ for $U'$ a submatrix of $U$ (so $U'$ could be $U$, or it could be a single entry, corresponding to your question).
This moment result in particular implies that $\sqrt{N}U_{1,1}$ tends to a standard complex Gaussian, see his Corollary 1.3.
However, although I couldn't find a precise reference, there is a shorter and easier proof, avoiding Weingarten calculus. I'll explain this in the case of the orthogonal group $O(N)$, which goes way back and for which I have references, and all the ideas adapt to $U(N)$.
In $O(N)$, the first column of a random Haar matrix is distributed like a random point on the sphere $\mathbb{S}^{N-1}$, and Borel studied the limiting behavior of the coordinates of such points already in 1906. A standard computation of the moments of these coordinates is given in Proposition 2.5 of Elizabeth Meckes' book ``The Random Matrix Theory of the Classical Compact Groups". It yields $$ \mathbb{E}_{O(N)} O_{1,1}^{2t} = \frac{\Gamma(t+\frac{1}{2})\Gamma(\frac{N}{2})}{\Gamma(t+\frac{N}{2})\Gamma(\frac{1}{2})}.$$