Evan Sorensen (UWM) -- Busemann functions and semi-infinite geodesics in a semi-discrete space скачать в хорошем качестве

Evan Sorensen (UWM) -- Busemann functions and semi-infinite geodesics in a semi-discrete space

In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift. Based on joint work with Timo Seppäläinen.