Mathematical Digest

This article describes new results concerning the geometry of Brownian motion. Brownian motion is a model that describes the hectic, random jostling of particles.  The likelihood that the particles' paths will cross is measured by "intersection exponents", which are of special interest to physicists because of their importance in understanding phase transitions.  New work concerning the intersection exponents has led to the discovery of a new kind of random process, called stochastic Loewner evolution, which may prove extremely useful in physics.  The work also settled a 1982 conjecture of fractal pioneer Benoit Mandelbrot.  He suggested that the length of the "frontier", or outer edge, of a Brownian path is proportional to the diameter of the frontier (that is, the longest distance across the frontier).  Just as Mandelbrot predicted, the ratio has now been shown to be 4/3.

--- Allyn Jackson