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Carlo Beenakker
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If you only want the maximum of a single column or a single row, then for large $d$ this is the maximum of $d$ i.i.d. Gaussians with zero mean and variance $1/d$.


The maximum of all matrix elements is more subtle:

This problem has been considered in the large-$d$ limit by T. Jiang, 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 $d\rightarrow\infty$.

Numerical simulations that show the convergence of the probability distribution of $W_n$ in the case of random orthogonal matrices are given in Uncertainty principles and ideal atomic decomposition.

Carlo Beenakker
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  • 448
  • 651