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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<\i> expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the -dimensional white noise. A key ingredient in our approach is a Lindeberg principle<\i> 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<\i> of statistical mechanics systems that are disorder relevant<\i>, 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.
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Francesco Caravenna, Rongfeng Sun, Nikos Zygouras, Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. 19 (2017), no. 1, pp. 1–65DOI 10.4171/JEMS/660