Free fields and related topics

The planar Gaussian free eld (GFF) has emerged in recent years as an object of central
importance in probability theory. It is conjectured, and in some cases proved, to arise as a universal
scaling limit of a broad range of models, including the Ginzburg|Landau interface model, the
height function associated to planar domino tilings and the dimer model, and the characteristic
polynomials of random matrices. A characterisation theorem for the GFF has recently been
developed as a tool for rigorously identifying such scaling limits. However, the conditions of
this theorem are non-trivial to check in many cases and there are several directions in which it
could be strengthened or extended that would have signicant consequences. One challenging
but achievable goal is to develop a characterisation theorem for the GFF on general Riemann
surfaces, where, for example, it could be applied quite directly to show convergence of dimer
height functions. Another (more ambitious) direction is to investigate to what extent the existing
\conformal invariance" assumption that characterises the planar GFF can be weakened. Part of
this PhD project will be to investigate one or both of these problems.
The (discrete) GFF also has a rich set of connections to other probabilistic objects, such as
simple random walks and spanning trees. One perspective on the Gaussian free eld is that it is
a spin system taking values in Euclidean space, and motivations from random matrix theory and
condensed matter physics have inspired work on hyperbolic free elds (HFF), where the Euclidean
space is replaced by hyperbolic space (or super-space). Surprisingly, many of the connections
enjoyed by the Gaussian free eld have analogues in these settings, and these connections have
played a role in settling longstanding conjectures. Currently this is best understood from the per-
spective of random walks: the relationship between the GFF and simple random walks GFF:SRW
has a hyperbolic analogue HFF:VRJP, where the VRJP is a self-reinforced random process that
rst appeared in the probability literature some time ago. The other part of this PhD project
will involve understanding the connections the GFF enjoys, and developing new analogues for the
HFF. A rst target is to develop a \loop soup" theory for the HFF that mirrors the powerful loop
soup point of view on the GFF that has been advocated by Werner and others.
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