## Institute for Mathematical Sciences

## Preprint ims95-9

** C. Bishop, P. Jones, R. Pemantle, and Y. Peres**
* The Dimension of the Brownian Frontier is Greater than 1*

Abstract: Consider a planar Brownian motion run for finite time. The frontier or ``outer boundary'' of the path is the boundary of
the unbounded component of the complement. Burdzy (1989)
showed that the frontier has infinite length. We improve this
by showing that the Hausdorff dimension of the frontier is
strictly greater than 1. (It has been conjectured that the
Brownian frontier has dimension $4/3$, but this is still
open.) The proof uses Jones's Traveling Salesman Theorem and
a self-similar tiling of the plane by fractal tiles known as
Gosper Islands.

View ims95-9 (PDF format)