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SUMMARY:Limits of self-similar graphs and criticality of the Abelian Sandp
 ile Model - Matter\, M (Universit de Genve)
DTSTART:20100729T163000Z
DTEND:20100729T164500Z
UID:TALK25659@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We consider covering sequences of (Schreier) graphs arising fr
 om self-similar actions by automorphisms of rooted trees. The projective l
 imit of such an inverse system corresponds to the action on the boundary o
 f the tree and its connected components are the (infinite) orbital Schreie
 r graphs of the action. They can be approximated by finite rooted graphs u
 sing Hausdorff-Gromov convergence. An interesting example is given by the 
 Basilica group acting by automorphisms on the binary rooted tree in a self
 -similar fashion. We give a topological as well as a measure-theoretical d
 escription of the orbital limit Schreier graphs. In particular\, it is sho
 wn that they are almost all one-ended with respect to the uniform distribu
 tion on the boundary of the tree. We study the statistical-physics Abelian
  Sandpile Model on such sequences of graphs. The main mathematical questio
 n about this model is to prove its criticality -- the correlation between 
 sites situated far away each from the other is high -- what is typically d
 one by exhibiting\, asymptotically\, a power-law decay of various statisti
 cs. In spite of many numerical experiments\, the criticality of the model 
 was rigorously proven only in the case of the regular tree. We show that t
 he Abelian Sandpile Model on the limit Schreier graphs of the Basilica gro
 up is critical almost everywhere with respect to the uniform distribution 
 on the boundary of the tree.\n
LOCATION:Seminar Room 1\, Newton Institute
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