BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Loop-erased random walk on planar graphs - Ariel Yadin (Weizmann I
 nstitute)
DTSTART:20081118T140000Z
DTEND:20081118T150000Z
UID:TALK14299@talks.cam.ac.uk
CONTACT:Berestycki
DESCRIPTION:This talk focuses on loop-erased random walk\, or LERW. LERW i
 s a random self-avoiding curve obtained by erasing the loop in the traject
 ory of a random walk in chronological order. Lawler\, Schramm\, and Werner
  proved that LERW on the Euclidean plane converges to SLE(2) as the mesh g
 oes to 0. SLE\, or Schramm-Loewner Evolution\, is a fascinating random pro
 cess discovered by Oded Schramm in 1999. SLE arises as the scaling limit o
 f many models in mathematical physics. It has many wonderful properties\, 
 perhaps the most important is "conformal invariance". Lawler\, Schramm\, a
 nd Werner's proof for the scaling limit of LERW essentially uses the symme
 try of the lattice structure. The question arises whether a similar result
  holds even under perturbed lattices\; for example\, if only a small porti
 on of edges are removed from the original lattice. We extend Lawler\, Schr
 amm and Werner's result: For any planar Markov chain (that is a Markov cha
 in embedded into the complex plane so that edges do not cross one another)
 \, if the scaling limit of the Markov chain is planar Brownian motion\, th
 en the scaling limit of the loop erasure of the Markov chain is SLE(2). On
 e main example\, is loop-erased random walk on the super-critical percolat
 ion cluster\; that is\, the infinite component after super-critical percol
 ation on Z^2. Berger and Biskup showed that the random walk on the super-c
 ritical percolation cluster converges to Brownian motion. Thus\, our resul
 t implies that the loop-erased random walk on the super-critical percolati
 on cluster converges to SLE(2). Joint work with Amir Yehudayoff. 
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
END:VEVENT
END:VCALENDAR