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SUMMARY:Statistical mechanics on nonamenable graphs - Tom Hutchcroft (Camb
 ridge) 
DTSTART:20171121T161500Z
DTEND:20171121T171500Z
UID:TALK95959@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:Since the breakthrough works of Hara and Slade in the early 90
 ’s\, there has been a well-developed theory of mean-field criticality fo
 r statistical physics models in high-dimensional Euclidean space. This mea
 ns that critical models on these spaces are described by the same critical
  exponents as they are on\, say\, the 3-regular tree. While this is intuit
 ively due to the “expansiveness” of high-dimensional space\, the proof
 s are rather specific to the Euclidean setting. In 1996\, Benjamini and Sc
 hramm proposed a program of understanding percolation and other models on 
 arbitrary transitive graphs through their coarse geometric features\, such
  as their isoperimetry. Relatively little progress has been made however\,
  even under the presence of very strong geometric assumptions such as nona
 menability.\n\nIn the first half of the talk\, I will discuss the main pro
 blems and conjectures in the field. In the second half\, I will outline my
  recent work on special cases of these conjectures in which the graph has 
 certain special symmetry properties (namely\, a nonunimodular transitive s
 ubgroup of automorphisms). In particular\, I hope to be able to show a com
 plete proof that self-avoiding walk on the product T x Z of a 3-regular tr
 ee with the integers (an example of historical interest) has mean-field cr
 itical exponents.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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