Limits of self-similar graphs and criticality of the Abelian Sandpile Model
Seminar Room 1, Newton Institute
AbstractWe consider covering sequences of (Schreier) graphs arising from self-similar actions by automorphisms of rooted trees. The projective limit of such an inverse system corresponds to the action on the boundary of the tree and its connected components are the (infinite) orbital Schreier graphs of the action. They can be approximated by finite rooted graphs using 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 description of the orbital limit Schreier graphs. In particular, it is shown that they are almost all one-ended with respect to the uniform distribution on the boundary of the tree. We study the statistical-physics Abelian Sandpile Model on such sequences of graphs. The main mathematical question about this model is to prove its criticality -- the correlation between sites situated far away each from the other is high -- what is typically done by exhibiting, asymptotically, a power-law decay of various statistics. 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 the Abelian Sandpile Model on the limit Schreier graphs of the Basilica group is critical almost everywhere with respect to the uniform distribution on the boundary of the tree.
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