  This thesis is concerned with aspects of the integrable Temperley--Lieb loop
(TL($n$)) model on a vertically infinite lattice with two non-trivial
boundaries. When $n=1$ the ground state eigenvector of the transfer matrix of
this model can be interpreted as a probability distribution of the possible
states of the system. Because of special properties the transfer matrix has at
$n=1$, we can show that the eigenvector is a solution of the q-deformed
Knizhnik--Zamolodchikov equation, and we use this fact to explicitly calculate
some of the components of the eigenvector. In addition, recursive properties of
the transfer matrix allow us to compute the normalisation of the eigenvector,
and show that it is the product of four Weyl characters of the symplectic
group. The boundary condition of this model lends itself to calculations
relating to horizontal percolation. One of these calculations is a type of
correlation function that can be interpreted as the density of percolation
cluster crossings between the two boundaries of the lattice. It is an example
of a class of parafermionic observables recently introduced in an attempt to
rigorously prove conformal invariance of the scaling limit of critical
two-dimensional lattice models. We derive an exact expression for this
correlation function, and find that it can be expressed in terms of the same
symplectic characters as the normalisation. In order to better understand these
solutions, we use Sklyanin's scheme to perform separation of variables on the
symplectic character, transforming the multivariate character into a product of
single variable polynomials. Analysing the asymptotics of these polynomials
will lead, via the inverse transformation, to the asymptotic limit of the
symplectic character, and thus to the asymptotic limit of the ground state
normalisation and correlation function of the loop model.
