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.
"Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation." Electron. J. Probab. 23 1 - 26, 2018. https://doi.org/10.1214/18-EJP157