  Inspired by recent work of Alberts, Khanin and Quastel, we formulate general
conditions ensuring that a sequence of multi-linear polynomials of independent
random variables (called polynomial chaos expansions) converges to a limiting
random variable, given by a Wiener chaos expansion over the d-dimensional white
noise. A key ingredient in our approach is a Lindeberg principle for polynomial
chaos expansions, which extends earlier work of Mossel, O'Donnell and
Oleszkiewicz. These results provide a unified framework to study the continuum
and weak disorder scaling limits of statistical mechanics systems that are
disorder relevant, including the disordered pinning model, the (long-range)
directed polymer model in dimension 1+1, and the two-dimensional random field
Ising model. This gives a new perspective in the study of disorder relevance,
and leads to interesting new continuum models that warrant further studies.
