Exploring conformal invariance with hierarchical models

In the context of the AdS/CFT correspondence, in Euclidean signature, an important basic fact is the bijection between conformal transformations of the boundary and hyperbolic isometries of the bulk. An infinite regular tree with the graph distance can be seen as a quintessential bare-bones version of a hyperbolic space. It turns out there is a natural way to define analogues of conformal maps on the boundary of such a tree and, quite miraculously, these are in bijection with tree isometries. Moreover, a Euclidean QFT on this boundary is the same as a hierarchical model as considered by Dyson in his study of the long-range Ising model and by Wilson when he introduced the approximate renormalization group recursion. I will try to give a pedagogical introduction to this circle of ideas, and I will discuss a particular model where there is hope to be able to prove conformal invariance from first principles via a rigorous nonperturbative renormalization group approach.

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