Planar Ising model, bipartite dimers, and uniform spanning trees are classical examples of free fermionic lattice models in 2d. Given a sequence of large planar graphs carrying such a model with “mesh size going to zero”, one is interested in finding a relevant complex structure that describes the limit of correlation functions. In recent years, it has been observed that this description most naturally comes from special embeddings of planar graphs into R^{2,1} or R^{2,2}, the so-called s- and t-surfaces. The aim of the talk is to present recent convergence results obtained in this context for limits of planar random walks (joint work with Basok, Laslier, and Russkikh) and Ising model (joint work with Mahfouf and Park).