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BIRS Workshop Lecture Videos

Probability over random graphs and actions of free groups Austin, Timothy


Many of the classical models of statistical physics, such as the Ising and Potts models, can be defined over any underlying finite graph. The case of a sparse, randomly-generated underlying graph has received considerable recent attention from probabilists, largely guided by several far-reaching predictions about its behaviour from the physics literature.

When the underlying graph is chosen uniformly at random from all $(2d)$-regular graphs on $n$ vertices, and then $n$ is sent to infinity, the local neighbourhoods around most vertices look like larger and larger trees with high probability. This observation allows one to extract weak limit processes over an infinite $(2d)$-regular tree from sequences of models built over the finite graphs. That infinite tree can be viewed as the Cayley graph of a free group, and the limit process becomes a probability-preserving action of that group on a shift-space.

This point of view is the basis for various asymptotic analyses of probabilistic features of the finite models, and also for the definition of sofic entropy for free-group actions in ergodic theory. This talk will be a gentle introduction to these two fields and the connections between them.

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