Spin systems with hyperbolic symmetry: a survey

  • Roland Bauerschmidt

    DPMMS, University of Cambridge, Cambridge CB3 0WB, UK
  • Tyler Helmuth

    Mathematical Sciences, Durham University, Durham DH1 3LE, UK
Spin systems with hyperbolic symmetry: a survey cover
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Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal–insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the many open questions that remain.