Possible topics for postgraduate research

Convex hulls of stochastic processes

Suppose that a planar random walk takes $n$ steps, and we construct the convex hull of the trajectory, that is, the smallest convex polygon containing all the sites visited by the walk. Of interest are the large-$n$ asymptotics of various statistics associated with the convex hull, such as the number of faces, the perimeter length, and the area.

See here, here and here for recent work on this problem.

Several interesting problems remain, including some aspects of:

• Processes with heavy-tailed increments.
• Diffusions.
• Spaces other than Euclidean space.

Contact: Andrew Wade.