A number of two-dimensional physical systems studied in statistical mechanics and polymer chemistry are well-described by a lattice model. These models are often conjectured to possess conformally invariant scaling limits which may be used to make exact predictions for critical exponents describing the qualitative behaviour of the system. In this talk, we introduce a particular lattice model known as the discrete excursion, and describe its conformally invariant scaling limit. We then briefly mention an application relating certain functionals of loop-erased random walk to Brownian motion. For further details, please read Chapter 1.
Chapter 1 gives a detailed introduction, and (I think!) puts everything into context.
The first page of each chapter summarizes that chapter's contents.
Section 2.7: The excursion Poisson kernel is an important object and is used throughout.
Section 4.4 constructs interior-to-boundary excursion measure and is used in Section 4.5 to construct boundary-to-boundary excursion measure.
The principal theorem is stated in Section 5.5.
Loop-erased random walk and Fomin's identity are reviewed in Sections 6.2 and 6.3.
Theorem 6.4.2 resolves Fomin's conjecture (which is recalled in Section 6.1).