| I propose to investigate the continuum scaling limit of percolation models in two dimensions, a procedure used to formulate a macroscopic description of models constructed on a lattice by sending to zero the mesh of the lattice while focusing on macroscopic features of the model, enabling the study of the fractal nature and properties of conformal invariance of statistical mechanics models at a critical point. In particular, in two dimensions the scaling limit picture has deep connections with the theory of two-dimensional conformal fields developed by physicists, whose main goal is to give a complete classification of critical points. Percolation is a model for the random coloring of a lattice; it combines geometry and probability in a way that has attracted much attention from bot hmathematicians and physicists. Its choice is dictated by three main reasons: (1) the intrinsic interest of the model, (2) its role as a tool in the study and as a paradigm for the behavior of other models, and (3) the recent introduction of the Stochastic Loewner Evolution (SLE), a very powerful tool for the study of two-dimensional scaling limits in general and percolation scaling limits in particular. The main tools involved in the proposed research are probability theory and the theory of conformal maps, which come together in the study of SLE, currently one of the most exciting and rapidly growing fields in probability theory. Two-dimensional percolation also has a well-developed theory, and the recent progress in understanding its scaling limit has resulted in a revival of interest. The main goals of the proposed research are to deepen our understanding of scaling limits and the tools involved in their study, to apply those tools to models that have not yet been analyzed, and to address some of the motivating questions coming from statistical mechanics. |
