Lectures on the Spin and Loop $O(n)$ Models

The classical spin $O(n)$ model is a model on a $d$-dimensional lattice in which a vector on the $(n-1)$-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model ($n=1$), the XY model ($n=2$) and the Heisenberg model ($n=3$). We discuss questions of long-range order and decay of correlations in the spin $O(n)$ model for different combinations of the lattice dimension $d$ and the number of spin components $n$. The loop $O(n)$ model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight $n\ge0$ and an edge weight $x\ge 0$. Special cases include self-avoiding walk ($n=0$), the Ising model ($n=1$), critical percolation ($n=x=1$), dimer model ($n=1,x=\infty$), proper $4$-coloring ($n=2, x=\infty)$, integer-valued ($n=2$) and tree-valued (integer $n>=3$) Lipschitz functions and the hard hexagon model ($n=\infty$). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin $O(n)$ model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.