RANDOMNESS OF EIGENVECTORS GENERATED FROM NETWORKS WITH RANDOM TOPOLOGIES

A model for the generation of neural connections at birth led to the study of W, a random, symmetric, nonnegative definite linear operator defined on a finite, but very large, dimensional Euclidean space. A limit law, as the dimension increases, on the eigenvalue spectrum of W was proven, implying that realizations of W (being identified with organisms in a species) appear totally different on the microscopic level and yet have almost identical spectral densities. Evidence is given to suport the conjecture that, contrary to the deterministic aspect of the eigenvalues, the eigenvectors behave in a completely chaotic manner, which is described in terms of the normalized uniform (Haar) measure on the group of orthogonal transformations on a finite dimensional space. The validity of the conjecture would imply a tabula rasa property on the ensemble ( ″species″ ) of all realizations of W.