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SUMMARY:Empirical measures\, geodesic lengths\, and a variational formula 
 in first-passage percolation - Erik Bates\, University of Wisconsin-Madiso
 n 
DTSTART:20211102T140000Z
DTEND:20211102T150000Z
UID:TALK164056@talks.cam.ac.uk
CONTACT:Sourav Sarkar
DESCRIPTION:We consider the standard first-passage percolation model on Z^
 d\, in which each edge is assigned an i.i.d. nonnegative weight\, and the 
 passage time between any two points is the smallest total weight of a near
 est-neighbor path between them.  This induces a random ``disordered” geo
 metry on the lattice.  Our primary interest is in the empirical measures o
 f edge-weights observed along geodesics in this geometry\, say from 0 to [
 n\\xi]\, where \\xi is a fixed unit vector. For various dense families of 
 edge-weight distributions\, we prove that these measures converge weakly t
 o a deterministic limit as n tends to infinity. The key tool is a new vari
 ational formula for the time constant. In this talk\, I will derive this f
 ormula and discuss its implications for the convergence of both empirical 
 measures and lengths of geodesics.
LOCATION:MR12  Centre for Mathematical Sciences
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