3,600 research outputs found

### Tiles and colors

Tiling models are classical statistical models in which different geometric
shapes, the tiles, are packed together such that they cover space completely.
In this paper we discuss a class of two-dimensional tiling models in which the
tiles are rectangles and isosceles triangles. Some of these models have been
solved recently by means of Bethe Ansatz. We discuss the question why only
these models in a larger family are solvable, and we search for the Yang-Baxter
structure behind their integrablity. In this quest we find the Bethe Ansatz
solution of the problem of coloring the edges of the square lattice in four
colors, such that edges of the same color never meet in the same vertex.Comment: 18 pages, 3 figures (in 5 eps files

### Critical behaviour of the dilute O(n), Izergin-Korepin and dilute $A_L$ face models: Bulk properties

The analytic, nonlinear integral equation approach is used to calculate the
finite-size corrections to the transfer matrix eigen-spectra of the critical
dilute O(n) model on the square periodic lattice. The resulting bulk conformal
weights extend previous exact results obtained in the honeycomb limit and
include the negative spectral parameter regimes. The results give the operator
content of the 19-vertex Izergin-Korepin model along with the conformal weights
of the dilute $A_L$ face models in all four regimes.Comment: 23 pages, no ps figures, latex file, to appear in NP

### Triangular Trimers on the Triangular Lattice: an Exact Solution

A model is presented consisting of triangular trimers on the triangular
lattice. In analogy to the dimer problem, these particles cover the lattice
completely without overlap. The model has a honeycomb structure of hexagonal
cells separated by rigid domain walls. The transfer matrix can be diagonalised
by a Bethe Ansatz with two types of particles. This leads two an exact
expression for the entropy on a two-dimensional subset of the parameter space.Comment: 4 pages, REVTeX, 5 EPS figure

### A Guide to Stochastic Loewner Evolution and its Applications

This article is meant to serve as a guide to recent developments in the study
of the scaling limit of critical models. These new developments were made
possible through the definition of the Stochastic Loewner Evolution (SLE) by
Oded Schramm. This article opens with a discussion of Loewner's method,
explaining how this method can be used to describe families of random curves.
Then we define SLE and discuss some of its properties. We also explain how the
connection can be made between SLE and the discrete models whose scaling limits
it describes, or is believed to describe. Finally, we have included a
discussion of results that were obtained from SLE computations. Some explicit
proofs are presented as typical examples of such computations. To understand
SLE sufficient knowledge of conformal mapping theory and stochastic calculus is
required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to
the text and improved hyperref suppor

### Bethe Ansatz solution of triangular trimers on the triangular lattice

Details are presented of a recently announced exact solution of a model
consisting of triangular trimers covering the triangular lattice. The solution
involves a coordinate Bethe Ansatz with two kinds of particles. It is similar
to that of the square-triangle random tiling model, due to M. Widom and P. A.
Kalugin. The connection of the trimer model with related solvable models is
discussed.Comment: 33 pages, LaTeX2e, 13 EPS figures, PSFra

### Exact conjectured expressions for correlations in the dense O$(1)$ loop model on cylinders

We present conjectured exact expressions for two types of correlations in the
dense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodic
boundary conditions. These are the probability that a point is surrounded by
$m$ loops and the probability that $k$ consecutive points on a row are on the
same or on different loops. The dense O$(n=1)$ loop model is equivalent to the
bond percolation model at the critical point. The former probability can be
interpreted in terms of the bond percolation problem as giving the probability
that a vertex is on a cluster that is surrounded by \floor{m/2} clusters and
\floor{(m+1)/2} dual clusters. The conjectured expression for this
probability involves a binomial determinant that is known to give weighted
enumerations of cyclically symmetric plane partitions and also of certain types
of families of nonintersecting lattice paths. By applying Coulomb gas methods
to the dense O$(n=1)$ loop model, we obtain new conjectures for the asymptotics
of this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA

### The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the ground state

We consider the integrable dilute Temperley-Lieb (dTL) O($n=1$) loop model on
a semi-infinite strip of finite width $L$. In the analogy with the
Temperley-Lieb (TL) O($n=1$) loop model the ground state eigenvector of the
transfer matrix is studied by means of a set of $q$-difference equations,
sometimes called the $q$KZ equations. We compute some ground state components
of the transfer matrix of the dTL model, and show that all ground state
components can be recovered for arbitrary $L$ using the $q$KZ equation and
certain recurrence relation. The computations are done for generic open
boundary conditions.Comment: 25 pages, 30 figures, Updated versio

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