Polynomial chaos and scaling limits of disordered systems

  • Francesco Caravenna

    Università degli Studi di Milano-Bicocca, Italy
  • Rongfeng Sun

    National University of Singapore, Singapore
  • Nikos Zygouras

    Warwick University, Coventry, UK


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 -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.

Cite this article

Francesco Caravenna, Rongfeng Sun, Nikos Zygouras, Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. 19 (2017), no. 1, pp. 1–65

DOI 10.4171/JEMS/660