Another probabilistic construction of ${\mathrm{\Phi }}^{2n}$ in dimension 2

The main input of this note is to provide an alternative probabilistic approach to the ${\mathrm{\Phi }}^{2n}$ theory in dimension 2, based on concentration phenomenon of martingales associated to polynomials of Gaussian variables. This is based on an adaptation of the work of Lacoin-Rhodes-Vargas, in which exponential potentials associated to quantum Mabuchi K-energy are studied.