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 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. 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

Gastrointestinal stromal tumor

<p>Abstract</p> <p>Background</p> <p>GISTs are a subset of mesenchymal tumors and represent the most common mesenchymal neoplasms of GI tract. However, GIST is a recently recognized tumor entity and the literature on these stromal tumors has rapidly expanded.</p> <p>Methods</p> <p>An extensive review of the literature was carried out in both online medical journals and through Athens University Medical library. An extensive literature search for papers published up to 2009 was performed, using as key words, GIST, Cajal's cells, treatment, Imatinib, KIT, review of each study were conducted, and data were abstracted.</p> <p>Results</p> <p>GIST has recently been suggested that is originated from the multipotential mesenchymal stem cells. It is estimated that the incidence of GIST is approximately 10-20 per million people, per year.</p> <p>Conclusion</p> <p>The clinical presentation of GIST is variable but the most usual symptoms include the presence of a mass or bleeding. Surgical resection of the local disease is the mainstay therapy. However, therapeutic agents, such as Imatinib have now been approved for the treatment of advanced GISTs and others, such as everolimus, rapamycin, heat shock protein 90 and IGF are in trial stage demonstrate promising results for the management of GISTs.</p