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SUMMARY:Near-critical scaling limits - Christophe Garban (CNRS\, Lyon)
DTSTART:20101019T153000Z
DTEND:20101019T163000Z
UID:TALK26752@talks.cam.ac.uk
CONTACT:Berestycki
DESCRIPTION: Consider percolation on the triangular grid in the plane. The
  scaling limit (as n goes to infinity) of a critical rescaled percolation 
 \\omega_{p_c}^n \non the triangular grid of mesh 1/n is a very rich probab
 ilistic object (it is the analog of the Brownian motion for the Random wal
 k)\, and is now well understood thanks to the works of Smirnov and Schramm
 .\n\nThe purpose of a joint work with Gabor Pete and Oded Schramm is to ``
 visualize'' the phase transition which occurs at p_c\, from the perspectiv
 e of the scaling limit. A first natural attempt in this direction is to co
 nsider the scaling limit as n goes to infinity of non-critical rescaled pe
 rcolations \\omega_p^n (with p \\neq p_c). Unfortunately\, such scaling li
 mits are "degenerate". In order to obtain non-trivial "off-critical" scali
 ng limits\, $p$ and $n$ need to be rescaled accordingly.\n\nI will describ
 e in this talk what the natural renormalization is if one wishes to observ
 e the emergence of an infinite cluster\, seen from the continuous. The mai
 n result in joint work with G. Pete and O. Schramm is that "up to scaling"
 \, there is a unique near-critical scaling limit. This near-critical limit
  is not conformally invariant anymore\, but one can nevertheless give a pr
 ecise description of its "conformal defect".\n\nFinally\, I will discuss s
 ome new results about dynamical and near-critical regimes in the case of d
 ependent models like the Ising model or the Random-Cluster model. \n\nhttp
 ://www.umpa.ens-lyon.fr/~cgarban/
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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