Convergence of loop-erased random walk to SLE(2) in the natural parametrization (10rit143)


(California Institute of Technology)

(University of Regina)

Robert Masson ()


The Banff International Research Station will host the "Convergence of loop-erased random walk to SLE(2) in the natural parametrization" workshop from January 17 to 24, 2010.

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. For instance, water changes phase from solid to liquid at 0 C and liquid to gas at 100 C. In elaborate continuous physical systems it is useful to approximate this continuous system by a discrete, or lattice, model. These lattice models lend themselves better to simulation.

The introduction in 1999 of a collection of planar random curves called the Schramm-Loewner evolution (SLE) has been of fundamental importance in the field of statistical mechanics, as it has provided scaling limits for many extensively studied planar lattice models. We study one such model, the loop-erased random walk (LERW). While the LERW has been proved to scale to SLE, this scaling only takes into account the geometry of the path and ignores how the model evolves in time. We attempt to resolve this issue by proving the convergence of LERW to SLE as time-parametrized curves.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologia (CONACYT).