Polynomial chaos and scaling limits of disordered systems

Abstract

Abstract. Inspired by recent work of Alberts, Khanin and Quastel [AKQ14a], 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 [MOO10]. 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.

On the Moments of the $$(2+1)$$-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

“…Let us mention some related work on the directed polymer model on the hierarchical lattice. In particular, for the marginally relevant case, Alberts, Clark and Kocić in [ACK17] established the existence of a phase transition, similar to [CSZ17a]. And more recently, Clark [Cla17] computed the moments of the partition function around a critical window for the case of bond disorder.…”

The Dickman subordinator, renewal theorems, and disordered systems

“…Proof of relations (3.5) and (3.18). We recall the polynomial chaos expansion used in [CSZ17a,CSZ17b]. Let us introduce the random variables…”

“…In [CSZ17a] we proposed a different approach to study disorder relevance: when a model is disorder relevant, it should be possible to suitably rescale the disorder strength to zero, as the system size diverges, and still obtain a non-trivial limiting model where disorder is present. Such an intermediate disorder regime had been investigated in [AKQ14a,AKQ14b] for the directed polymer model based on the simple random walk on Z, which is disorder relevant.…”

“…where we work with X 2n rather than X n to avoid periodicity issues. Writing Z β N as a polynomial chaos expansion [CSZ17a] (we review the computation in Appendix A.1), we obtain the following expression for the second moment:…”

Approximation to stable law by the Lindeberg principle

“…al. [8,7] obtained a general scaling of disordered system by expanding polynomial chaos. Besides a lot of applications in random matrices [4,6,26,41,29,42,1,5], Lindeberg principle has also been applied to other research 1 areas such as high dimensional regressions [14,15,20], time series [21,41,32,31], bootstrap [35,15], statistical learning [27,44] and so on.…”