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SUMMARY:Ken Alexander (USC)
DTSTART:20200501T160000Z
DTEND:20200501T170000Z
DTSTAMP:20260423T005734Z
UID:PatC/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PatC/3/">Geo
 desics\, bigeodesics\, and coalescence in first passage percolation in gen
 eral dimension</a>\nby Ken Alexander (USC) as part of Probability and the 
 City Seminar\n\n\nAbstract\nIn first passage percolation (FPP) on $\\mathb
 b{Z}^d$\, i.i.d.~(bond) passage times are attached to the nearest-neighbor
  bonds of the lattice\, and the passage time from $x$ to $y$ is the shorte
 st sum of bond passage times among all possible paths from $x$ to $y$\; th
 e corresponding minimizing path is called a geodesic. One can also conside
 r geodesic rays and bigeodesics\, which are one-ended and two-ended infini
 te paths for which every finite segment is a geodesic\; a $\\theta$-ray is
  a geodesic ray with asymptotic direction $\\theta$. The conjectured pictu
 re\, partly verified for $d=2$ under assumptions of various strengths\, is
  that for a given $\\theta$\, there is a.s.~a unique $\\theta$--ray from e
 ach lattice site\, and any two $\\theta$--rays eventually coalesce\, thoug
 h there is a random null set of directions for which this fails\; bigeodes
 ics a.s.~do not exist at all. Here we establish portions of this heuristic
  picture in higher dimensions (where few results currently exist)\, at lea
 st under the assumption that certain very basic but unproven properties of
  FPP are valid. We establish a coalescence-like property: taking all the $
 \\theta$--rays starting next to a given hyperplane\, and looking at the se
 t of points where they cross another hyperplane some distance $r$ ahead of
  the starting one\, we show that the geodesics bundle together in the sens
 e that the density of the crossing points approaches 0 (at a near-sharp ra
 te) as $r\\to\\infty$. This bundling property also holds if we consider to
 gether all $\\theta$--rays over a narrow range of directions $\\theta$\, a
 nd this fact leads to proof of the absence of bigeodesics. In $d=2$\, bund
 ling can be used to bound the probability that two $\\theta$--rays do not 
 coalesce before traveling a distance $r$.\n
LOCATION:https://researchseminars.org/talk/PatC/3/
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