Random Banach spaces: The limitations of the method

We study the properties of "generic", in the sense of the Haar measure onthe corresponding Grassmann manifold, subspaces of l^N_infinity of givendimension. We prove that every "well bounded" operator on such a subspace, sayE, is a "small" perturbation of a multiple of identity, where "smallness" isdefined intrinsically in terms of the geometry of E. In the oppositedirection, we prove that such "generic subspaces of l^N_infinity" do admit"nontrivial well bounded" projections, which shows the "near optimality" ofthe first mentioned result, and proves the so called "Pisier's dichotomyconjecture" in the "generic" case.