Couplings of SLE and Gaussian free field:

There is a way to couple the Schramm-Loewner evolution SLE(kappa) with the Gaussian free field so that conditioned on the path, the expected height of the field near the path is determined by the winding of the path (see the Fields Institute lectures for a precise explanation) . The following figures represent discrete analogs of this coupling. To draw such a figure, one first samples a discrete Gaussian free field with appropriate boundary conditions---this gives us a random real-valued function h on the hexagonal faces (which are colored according to the function values). The black paths are then generated dynamically from bottom to top. When the path hits a hexagon it has not visited before, it decides whether to turn left or right by comparing a constant multiple of the winding of the path (number of right minus number of left turns thus far) with the value of h on that hexagon. Different constants correspond to different values of kappa.

When the constant is zero, the winding is irrelevant, and the path always turns right when h is negative and left when it is positive. In this case, the path is a boundary between clusters of positively and negatively valued hexagons. This case corresponds to kappa = 4 and is studied in detail in joint work with Oded Schramm.

A figure similar to those below appeared in Scientific American .