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

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.Comment: PhD Thesis, 130 pages, 126 figure

Finite-Size Left-Passage Probability in Percolation

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. In terms of clusters, this corresponds to the one-arm probability. Our calculation is based on the q-deformed Knizhnik-Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm's left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin-Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary term

Finite-size left-passage probability in percolation

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm's left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure

Exact finite size groundstate of the O(n=1) loop model with open boundaries

We explicitly describe certain components of the finite size groundstate of the inhomogeneous transfer matrix of the O(n=1) loop model on a strip with non-trivial boundaries on both sides. In addition we compute explicitly the groundstate normalisation which is given as a product of four symplectic characters.Comment: 29 pages, 33 eps figures, major revisio

Finite-size corrections for universal boundary entropy in bond percolation

We compute the boundary entropy for bond percolation on the square lattice in the presence of a boundary loop weight, and prove explicit and exact expressions on a strip and on a cylinder of size $L$. For the cylinder we provide a rigorous asymptotic analysis which allows for the computation of finite-size corrections to arbitrary order. For the strip we provide exact expressions that have been verified using high-precision numerical analysis. Our rigorous and exact results corroborate an argument based on conformal field theory, in particular concerning universal logarithmic corrections for the case of the strip due to the presence of corners in the geometry. We furthermore observe a crossover at a special value of the boundary loop weight