The Schramm-Loewner evolution and scaling limits of discrete planar processes

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One of the broad goals of statistical mechanics is to understand the behaviour of a physical system at criticality; that is, at (or near) the temperature at which a phase transition occurs. A well-known example of a phase transition occurs when water freezes (i.e., it changes state from liquid to solid) or when it boils (i.e., it changes state from solid to gas). In more elaborate models, the continuous physical system is well-described by a discrete, or lattice, model. These lattice models are often more tractable mathematically, and often physical or chemical predictions about the continuous system can be proved rigorously using results established for the lattice model. An important class of systems are two-dimensional; by first understanding two-dimensional models, we may better hope to understand our three-dimensional world. An exciting development occurred in 2000 when Oded Schramm introduced the stochastic Loewner evolution (SLE) while studying scaling limits of loop-erased random walk. SLE is a new class of conformally invariant stochastic processes which has revolutionized the study of critical phenomenon in statistical mechanics. In fact, its importance was punctuated by the awarding of the Fields Medal to Wendelin Werner "for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.'' Although this program of study has already produced several deeply significant results, there is still much more work to be done. For instance, the self-avoiding random walk, a model of polymer chains introduced by the Nobel prize-winning chemist Paul Flory in the 1940's, is a model where minimal mathematical progress has been made and is one of the motivations for much of this entire program of study. Another important area of study that has not yet received much attention is the case of multiple interfaces that occur naturally in, for example, ferromagnetism and the corresponding Ising model. Therefore, the primary goal of this research program is to study multiple SLE curves and their interactions which is the natural continuous object to study when considering multiple interfaces.