Let $U$ be a $d\times d$ unitary matrix, and $U_{i,j}$ be its matrix elements. I am interested in the following quantity $$\int dU \max_j |U_{1,j}|^2 \ , $$ where $dU$ is the uniform Haar measure over ${\rm SU}(d)$.

Please let me know if you have any idea for calculating this integral for general $d$.

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    $\begingroup$ Your title asks for the maximum matrix element (out of, presumably, all of them), but your integral only selects the maximum element in the first row. Which do you mean? $\endgroup$ – LSpice Dec 15 '19 at 2:15
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    $\begingroup$ I believe I've read that the joint distribution of the $|U_{1,j}|^2$ is the uniform distribution on the standard $d$-simplex. $\endgroup$ – MTyson Dec 15 '19 at 4:11

The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting:

$$\int dU \max_j |U_{1,j}|^2 =\frac{1}{d}\sum_{j=1}^d \frac{1}{j}.$$

For large $d$ this tends to $(1/d)\log d$. The complete probability distribution of the row-maximum is known.

The maximum of all matrix elements is more difficult and only large-$d$ asymptotics has a closed-form expression, see Maxima of entries of Haar distributed matrices (alternative link).

If $W_d$ is the maximum matrix element in absolute value of a Haar-distributed $d\times d$ unitary matrix, then $W_d^2\rightarrow (2/d)\log d$ in probability for $d\rightarrow\infty$, so twice as large as for the row-maximum.

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$\newcommand{\C}{\mathbb C}$$\newcommand{\R}{\mathbb R}$Any linear isometry of $\C^d$ is a unitary transformation. Therefore, the distribution of the random vector $V:=(X_1,Y_1,\dots,X_d,Y_d)$ is uniform on the unit sphere in $\R^{2d}$, where $X_j:=\Re U_{1,j}$ and $Y_j:=\Im U_{1,j}$. So, $V$ equals $W:=(W_1,\dots,W_{2d})$ in distribution, where $$W_j:=\frac{Z_j}{\sqrt{Z_1^2+\dots+Z_{2d}^2}} $$ and $Z_1,\dots,Z_{2d}$ are iid standard normal random variables (r.v.'s). So, the random vector $(|U_{1,1}|^2 ,\dots,|U_{1,d}|^2)$ equals $R:=(R_1,\dots,R_d)$ in distribution, where $$R_j:=\frac{T_j}{T_1+\dots+T_d} $$ and $T_k:=Z_{2k-1}^2+Z_{2k}^2$, so that $T_1,\dots,T_d$ are iid standard exponential r.v.'s. So, $\max_j|U_{1,j}|^2$ equals $\max_j R_j$ in distribution. The distribution and, in particular, the expectation of $\max_j R_j$ were found a long time ago; see e.g. Irwin (1955) and historical references therein going back to as far as 1897. In particular, according to formula (15) in Cochran's paper, $$E\max_j|U_{1,j}|^2=E\max_j R_j=\frac1d\,\Big(1+\frac12+\dots+\frac1d\Big). $$

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