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 µ 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.
Powell, E. (2018). Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation. Electronic Journal of Probability, 23, 1-26. https://doi.org/10.1214/18-ejp157