Spectral properties of random non-self-adjoint matrices and operators

I describe some numerical experiments which determine the degree of spectra l instability of medium-sized randomly generated matrices which are far fro m self-adjoint. The conclusion is that the eigenvalues are likely to be int rinsically uncomputable for similar matrices of a larger size. I also descr ibe a stochastic family of bounded operators in infinite dimensions for alm ost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. My results imply that the spect rum of the non-self-adjoint Anderson model changes suddenly on passing to t he infinite volume limit.