Can one hear the shape of a random walk?

Stochastics Seminar
Thursday, September 29, 2016 - 15:05
1 hour (actually 50 minutes)
Skiles 006
Texas A&M University
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. This is joint work with Marek Biskup.