Random correlation matrices

Given a bijective vectorial Boolean function Z~-1-Z~, define the correlation matrix P as an N x N matrix, N = 2 n-1, whose entries are given as the squares of the correlation coefficients between nonzero linear combinations of the component Boolean functions of F and nonzero linear Boolean functions of the same n variables. Let A denote the number of nonzero entries in P. When F is chosen uniformly at random, the expected value and variance of A are determined. As a consequence, it is shown that for any 'YN = o(N), the fraction of all F such that A::S; N'YN is o(N-1). Similar results are also obtained for partially linear F. When F is such that (1-cn)n component functions of F are necessarily linear, where cnn-1-00 as n-1-00, it is derived that for any 'YN = o(N), the fraction of all F such that A ::s; N'YN is O(Nc n-2).