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This thesis is concerned with aspects of the integrable Temperley--Lieb loop(TL($n$)) model on a vertically infinite lattice with two non-trivialboundaries. When $n=1$ the ground state eigenvector of the transfer matrix ofthis model can be interpreted as a probability distribution of the possiblestates 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-deformedKnizhnik--Zamolodchikov equation, and we use this fact to explicitly calculatesome of the components of the eigenvector. In addition, recursive properties ofthe transfer matrix allow us to compute the normalisation of the eigenvector,and show that it is the product of four Weyl characters of the symplecticgroup. The boundary condition of this model lends itself to calculationsrelating to horizontal percolation. One of these calculations is a type ofcorrelation function that can be interpreted as the density of percolationcluster crossings between the two boundaries of the lattice. It is an exampleof a class of parafermionic observables recently introduced in an attempt torigorously prove conformal invariance of the scaling limit of criticaltwo-dimensional lattice models. We derive an exact expression for thiscorrelation function, and find that it can be expressed in terms of the samesymplectic characters as the normalisation. In order to better understand thesesolutions, we use Sklyanin's scheme to perform separation of variables on thesymplectic character, transforming the multivariate character into a product ofsingle variable polynomials. Analysing the asymptotics of these polynomialswill lead, via the inverse transformation, to the asymptotic limit of thesymplectic character, and thus to the asymptotic limit of the ground statenormalisation and correlation function of the loop model.

Finite size lattice results for the two-boundary Temperley--Lieb loop model