# Moments of the 2D SHE at criticality

## TL;DR

This paper investigates the critical behavior of the two-dimensional stochastic heat equation with multiplicative noise, establishing convergence of correlation functions to a limit described by an explicit semigroup, using resolvent analysis of the delta Bose gas.

## Contribution

It introduces a novel approach to analyze the critical limit of the 2D stochastic heat equation via resolvent convergence of the delta Bose gas and explicit semigroup characterization.

## Key findings

- Correlation functions converge to a non-trivial limit
- Explicit semigroup describes the critical behavior
- Method adapts resolvent analysis to mollified noise setup

## Abstract

We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius $ \varepsilon\to 0 $, we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit non-trivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of Dimock and Rajeev (2004) to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.