We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. These log-correlated non-Gaussian random fields arise in different branches of statistical mechanics. Existing results were mainly focused on the CLT for the linear functionals. In this talk I will describe some recent progress on the global maximum and local CLT for the field, thus confirming they are in the Gaussian universality class in a very strong sense. The proof uses a random walk representation (a la Helffer-Sjostrand) and an approximate harmonic coupling (by J. Miller).
