Department Colloquium

Department of Mathematics
Princeton University


November 15th, 2000

Scaling limits of random processes and the outer boundary of planar Brownian motion



Consider a random walk on the square grid in the plane: a particle is placed at the origin, and at each step moves to a random vertex adjacent to the current position, with all choices having equal probability. If one performs this process on finer and finer grids, and rescales time appropriately, the process converges to a random path, which is called Brownian motion. Perhaps surprisingly, Brownian motion is more symmetric than the random walk: it has rotational symmetry. In fact, it is conformally invariant, which is a more general kind of symmetry.

A more complicated process is critical percolation, where each edge of the square grid is deleted with probability 1/2, independently, and the connectivity properties of the resulting graph are studied.

It is an outstanding challenge to understand what happens to critical percolation and similar processes when the mesh of the grid tends to zero. Many of these processes are believed to display conformal invariance in the limit, but this is mostly unproven. Under the assumption of conformal invariance we give a complete description of the scaling limit of critical percolation and several other models. The description is based on a process that we call Stochastic Loewner Evolution (SLE). The SLE process describes a randomly growing set by specifying the conformal map to the complement of the set. The conformal map is obtained by solving a random differential equation. There's one free parameter $\kappa>0$ in the description of SLE.

In joint work with Greg Lawler and Wendelin Werner, we prove that many properties of planar Brownian Motion are the same as those of SLE with $\kappa=6$. This is then used to answer several problems regarding planar Brownian Motion. In particular, we prove Mandelbrot's conjecture stating that the Hausdorff dimension of the outer boundary of planar Brownian Motion is 4/3.

The talk will assume no prior specialized knowledge. The plan is to describe some of the random processes, explain and motivate the construction of SLE, and explain the relation with planar Brownian Motion.