An invariance principle for the two-dimensional parabolic Anderson model with small potential

TL;DR

This paper establishes an invariance principle for the 2D lattice parabolic Anderson model with small potential, leading to convergence results for random polymers and universality in spectral properties of discrete Schrödinger operators.

Contribution

It introduces a novel invariance principle for the 2D parabolic Anderson model with small potential, connecting it to random polymers and spectral universality.

Findings

Proves an invariance principle for the 2D parabolic Anderson model.

Derives a Donsker type convergence for discrete random polymers.

Establishes universality for the spectrum of discrete random Schrödinger operators.

Abstract

We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schr\"odinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some basic results for multiple stochastic integrals of discrete martingales.