Student Probability Seminar

Mixing planer vector field without asymptotic directions

Speaker: Liying Li, CIMS

Location: Warren Weaver Hall 1314

Date: Monday, March 25, 2019, 12:30 p.m.


Infinite geodesics and their asymptotic behavior is one of the key objects studied in last-passage percolation.  For many models, especially those that are known to have strictly convex shape functions, infinite geodesics always exist and have asymptotic directions, and is unique for every fixed direction and starting point.  For every fixed direction, if we consider the tangent vectors generated by all these geodesics, we obtain an ergodic vector field, which inherits the ergodicity from the random environment in the LPP process.

One can ask the reverse question: for an ergodic vector field, will its integral curves always have asymptotic directions?  In this talk, we will present an interesting counter-example of a strongly mixing stationary smooth planer vector field, whose integral curves have slopes oscillating between 0 and +\infty.  The construction is based on placing long corridors with random heavy-tail strengths and lengths at locations given by the 2D homogeneous Poissonian point process.