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CATEGORIES:Probability
SUMMARY:Near-critical scaling limits - Christophe Garban (
CNRS\, Lyon)
DTSTART;TZID=Europe/London:20101019T163000
DTEND;TZID=Europe/London:20101019T173000
UID:TALK26752AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/26752
DESCRIPTION: Consider percolation on the triangular grid in th
e 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 r
ich probabilistic object (it is the analog of the
Brownian motion for the Random walk)\, and is now
well understood thanks to the works of Smirnov and
Schramm.\n\nThe purpose of a joint work with Gabo
r Pete and Oded Schramm is to ``visualize'' the ph
ase transition which occurs at p_c\, from the pers
pective of the scaling limit. A first natural atte
mpt in this direction is to consider the scaling l
imit as n goes to infinity of non-critical rescale
d percolations \\omega_p^n (with p \\neq p_c). Unf
ortunately\, such scaling limits are "degenerate".
In order to obtain non-trivial "off-critical" sca
ling limits\, $p$ and $n$ need to be rescaled acco
rdingly.\n\nI will describe 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 main result in joint work wit
h G. Pete and O. Schramm is that "up to scaling"\,
there is a unique near-critical scaling limit. Th
is near-critical limit is not conformally invarian
t anymore\, but one can nevertheless give a precis
e description of its "conformal defect".\n\nFinall
y\, I will discuss some new results about dynamica
l and near-critical regimes in the case of depende
nt models like the Ising model or the Random-Clust
er model. \n\nhttp://www.umpa.ens-lyon.fr/~cgarban
/
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Berestycki
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