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Michael Damron: Absence of backward infinite paths in first-passage percolation in arbitrary dimension

Time: Fri 2019-11-29 14.30

Location: Kräftriket, house 6, room 306 (Cramér-rummet)

Participating: Michael Damron, Georgia Tech

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In first-passage percolation (FPP), one places weights \((t_e)\) on the edges of \(\mathbb{Z}^d\) 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).