  We study the stability of the Lanczos algorithm run on problems whose
eigenvector empirical spectral distribution is near to a reference measure with
well-behaved orthogonal polynomials. We give a backwards stability result which
can be upgraded to a forward stability result when the reference measure has a
density supported on a single interval with square root behavior at the
endpoints. Our analysis implies the Lanczos algorithm run on many large random
matrix models is in fact forward stable, and hence nearly deterministic, even
when computations are carried out in finite precision arithmetic. Since the
Lanczos algorithm is not forward stable in general, this provides yet another
example of the fact that random matrices are far from "any old matrix", and
care must be taken when using them to test numerical algorithms.
