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**Graphical approach to lattice spin models**
Hugo Duminil-Copin, in PIMS summer school in Probability
(UBC), 2017.

**Abstract.** Phase transitions
are a central theme of statistical mechanics, and of
probability more generally. Lattice spin models
represent a general paradigm for phase transitions in
finite dimensions, describing ferromagnets and even some
fluids (lattice gases). It has been understood since the
1980s that random geometric representations, such as the
random walk and random current representations, are
powerful tools to understand spin models. In addition to
techniques intrinsic to spin models, such
representations provide access to rich ideas from
percolation theory. In recent years, for two-dimensional
spin models, these ideas have been further combined with
ideas from discrete complex analysis. Spectacular
results obtained through these connections include the
proof that interfaces of the two-dimensional Ising model
have conformally invariant scaling limits given by SLE
curves, that the connective constant of the
self-avoiding walk on the hexagonal lattice is given by
$\sqrt{2+\sqrt 2}$, and that the magnetisation of the
three-dimensional Ising model vanishes at the critical
point.