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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 (Gumbel distribution).

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.

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.

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:

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

with $H_d=\sum_{j=1}^d 1/j$ the harmonic number. For large $d$ this tends to $(1/d)\log d$. The complete probability distribution of the row-maximum is known (Gumbel distribution).

The maximum of all matrix elements is more difficult and only large-$d$ asymptotics is known, 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.

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 (Gumbel distribution).

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.

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:

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

with $H_d=\sum_{j=1}^d 1/j$ the harmonic number. For large $d$ this tends to $(1/d)\log d$.

The maximum of all matrix elements is more subtle:

This problem has been considered in the large-$d$ limit in 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.

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:

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

with $H_d=\sum_{j=1}^d 1/j$ the harmonic number. For large $d$ this tends to $(1/d)\log d$. The complete probability distribution of the row-maximum is known (Gumbel distribution).

The maximum of all matrix elements is more difficult and only large-$d$ asymptotics is known, 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.