Maxima of entries of Haar distributed matrices

Let _n=(_ij) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W_n:=max_1i,jn|_ij|. We obtain the limiting distribution and a strong limit theorem on W_n. A tool has been developed to prove these results. It says that up to n/( log n)² columns of _n can be approximated simultaneously by those of some Y_n=(y_ij) in which y_ij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n).Mathematics Subject Classification (2000):15A52, 60B10, 60B15, 60F10