Below are Gif, PostScript and Data Files of pictures coming from the Loewner differential equation from univalent function theory with Brownian motion as a driving term. See Oded Schramm's paper ``Scaling limits of loop-erased random walks and uniform spanning trees'' (Israel Jour. Math. 118 (2000), 221--288) for background and motivation. The DataFiles are best viewed with the program xcm obtainable from Don Marshalls website. The advantage over the Gif or PostScript file is that you can continuously zoom into the picture.

The pictures below are named p k_n. They are approximations to the solution of the Loewner differential equation with parameter B(k t), where B(t) is Brownian motion (and t ranges from 0 to 1). The pictures represent solutions to Loewners equation with B being sampled at n equidistributed points.  The image of n points under the time 1 map is calculated, and these n points are joined by line segments. See Schramm's paper for the (conjectural) relevance of the parameters k=2,4 and 8.

The algorithm leading to the pictures was obtained in joint work with Don Marshall. A manuscript of our work is in preparation. The program creating the pictures was written by our undergradute research assistants Tarn Adams, Gary Look and Julie Rowlett.
 

Gif                        Data                  PostScript

p2_10000.gif       p2_10000        p2_10000.ps
p2_20000.gif       p2_20000        p2_20000.ps
p2b_20000.gif       p2b_20000       p2b_20000.ps
p2_50000.gif       p2_50000        p2_50000.ps
p4_10000.gif       p4_10000       p4_10000.ps
p4b_10000.gif      p4b_10000      p4b_10000.ps
p4_45000.gif       p4_45000       p4_45000.ps
p4_65000.gif       p4_65000       p4_65000.ps
p8_10000.gif       p8_10000       p8_10000.ps