# Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

## TL;DR

This paper introduces a discrete paracontrolled distribution framework on Bravais lattices, combining it with a martingale approach to establish weak universality of the 2D parabolic Anderson model, showing it as a scaling limit of nonlinear models.

## Contribution

It develops a novel discrete paracontrolled distribution method and a systematic martingale approach for analyzing scaling limits of lattice systems.

## Key findings

- Established a discrete paracontrolled distribution framework on Bravais lattices.
- Proved weak universality of the 2D parabolic Anderson model as a scaling limit.
- Demonstrated the approach's effectiveness in controlling moments of i.i.d. polynomial functions.

## Abstract

We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distribution in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model.