image

25 / Schramm-Loewner evolution for different values of k. For example, critical percolation (top) gives trajectories with κ = 6. The middle image shows κ = 0.5, and the bottom image κ = 8/3. Reprinted with permission from Scott Sheffield, Massachusetts Institute of Technology. /

from measures of the boundaries between small regions of materials (“magnetic grains”) that are aligned magnetically (κ = 3) or paths for water percolation (κ = 6) (Figure 25). Schramm-Loewner evolution is a wonderful unified description of all these separate phenomena, illuminating how disorder can create order.

The most essential feature of Schramm-Loewner evolution is a symmetry property called conformal invariance. Conformal invariance has two components: scale invariance and rotation invariance. The first means that a Brownian trajectory will look just the same at any level of magnification. If you blow it up by a factor of 10, it will look just as jiggly as before. The formerly small bounces will become big bounces—but you will see new, even smaller bounces that you couldn’t make out before. Rotation invariance means that Brownian motion has no preferred direction. For example, in a closed room, smoke will go everywhere.

Consider a crack that grows randomly inward from the edge of an infinite pane of glass. According to the idea on which Schramm-Loewner evolution is based, the pane of glass can always be “healed” by a conformal transformation, deforming the glass in a way that pushes the crack back out to the boundary. If the crack grows, the glass can be healed again. In this never-ending process of cracking and healing, the attachment point of the crack moves around. In fact, it jiggles very erratically along the edge of the glass. Does this sound familiar? The attachment point is actually undergoing one-dimensional Brownian motion. The intensity of the jiggling is described by the parameter κ larger values of κ correspond to more intense jiggling and to more jagged cracks. The basic idea of Schramm-Loewner evolution converts any two-dimensional, conformally invariant random process into a one-dimensional Brownian motion. Many questions become simpler after they are restated as a Schramm-Loewner evolution. What is the probability that the trajectories of two pollen grains in a petri dish will intersect before the grains reach the edge of the dish? What is the probability that water will percolate from one side of a



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement