# Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation

## TL;DR

This paper proves the convergence and uniqueness of critical Gaussian chaos measures with derivative normalisation for a broad class of log-correlated fields, including the 2D Gaussian free field, showing independence from approximation methods.

## Contribution

It establishes the convergence and uniqueness of the critical chaos measure with derivative normalisation for general convolution approximations of log-correlated fields.

## Key findings

- Critical chaos measures converge to a unique limit.
- The limiting measure is independent of the approximation method.
- The measure matches those obtained via Seneta–Heyde renormalisation and white-noise approximation.

## Abstract

We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure {\mu}'. This limiting measure does not depend on the choice of approximation. Moreover, it is equal to the measure obtained using the Seneta--Heyde renormalisation at criticality, or using a white-noise approximation to the field.