# Average of the maximum matrix element over the Haar measure

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 $$d$$.

• 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? – LSpice Dec 15 '19 at 2:15
• I believe I've read that the joint distribution of the $|U_{1,j}|^2$ is the uniform distribution on the standard $d$-simplex. – MTyson Dec 15 '19 at 4:11

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