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CATEGORIES:Probability
SUMMARY:Statistical mechanics on nonamenable graphs - Tom
Hutchcroft (Cambridge)
DTSTART;TZID=Europe/London:20171121T161500
DTEND;TZID=Europe/London:20171121T171500
UID:TALK95959AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/95959
DESCRIPTION:Since the breakthrough works of Hara and Slade in
the early 90’s\, there has been a well-developed t
heory of mean-field criticality for statistical ph
ysics models in high-dimensional Euclidean space.
This means that critical models on these spaces ar
e described by the same critical exponents as they
are on\, say\, the 3-regular tree. While this is
intuitively due to the “expansiveness” of high-dim
ensional space\, the proofs are rather specific to
the Euclidean setting. In 1996\, Benjamini and Sc
hramm proposed a program of understanding percolat
ion and other models on arbitrary transitive graph
s through their coarse geometric features\, such a
s their isoperimetry. Relatively little progress h
as been made however\, even under the presence of
very strong geometric assumptions such as nonamena
bility.\n\nIn the first half of the talk\, I will
discuss the main problems and conjectures in the f
ield. In the second half\, I will outline my recen
t work on special cases of these conjectures in wh
ich the graph has certain special symmetry propert
ies (namely\, a nonunimodular transitive subgroup
of automorphisms). In particular\, I hope to be ab
le to show a complete proof that self-avoiding wal
k on the product T x Z of a 3-regular tree with th
e integers (an example of historical interest) has
mean-field critical exponents.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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