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Probability and Combinatorics

Tuesday, November 5, 2019 - 3:00pm

Michael Damron

Georgia Tech

Location

Temple University

617 Wachmann Hall

In first-passage percolation (FPP), one places weights (t_e) 
on the edges of Z^d and considers the induced metric. Optimizing paths 
for this metric are called geodesics, and infinite geodesics are 
infinite paths all whose finite subpaths are geodesics. It is a major 
open problem to show that in two dimensions, with i.i.d. continuous 
weights, there are no bigeodesics (doubly-infinite geodesics). In this 
talk, I will describe work on bigeodesics in arbitrary dimension using 
``geodesic graph'' measures introduced in '13 in joint work with J. 
Hanson. Our main result is that these measures are supported on graphs 
with no doubly-infinite paths, and this implies that bigeodesics cannot 
be constructed in a translation-invariant manner in any dimension as 
limits of point-to-hyperplane geodesics. Because all previous works on 
bigeodesics were for two dimensions and heavily used planarity and 
coalescence, we must develop new tools based on the mass transport 
principle. Joint with G. Brito (Georgia Tech) and J. Hanson (CUNY).