# Expected even power of absolute value of an element of a random unitary matrix

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.

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 $$\mathbb{E}_{O(N)} O_{1,1}^{2t} = \frac{\Gamma(t+\frac{1}{2})\Gamma(\frac{N}{2})}{\Gamma(t+\frac{N}{2})\Gamma(\frac{1}{2})}.$$
• Thanks! Do you know if there's also a formula for $\mathbb{E}|U_{1,2}|^{2t}$? Nov 2, 2021 at 13:41
• Haar measure is invariant under multiplication by a unitary matrix, in particular by permutation matrices. This implies that all $U_{i,j}$ follow the same law. This is also covered in chapter 2 of E. Meckes' book. Nov 2, 2021 at 16:02