This is a note based on a talk1 by Oded Schramm in ICM(International Congress of Mathematicians) 2006, which features a quite intuitive approach to the idea of scaling limit appearing a lot in today’s math/physics literature. The illustration features mainly the case of triangular lattice. In this specific situation, thanks to the theorem of Smirnov2, we are able to characterize the behavior of the limit very well.
This particular discovery is one of the motivations at very beginning of the idea Schramm–Loewner evolution.
The note attached here is nothing new but all from the reference. As I made some effort on drawing the pictures step by step, I hope this will make you get some good ideas behind it.
Schramm: Conformally invariant scaling limits (an overview and a collection of problems), arXiv:math/0602151 ↩︎
S. Smirnov: Critical percolation in the plane, arXiv:0909.4499 ↩︎