23,507 research outputs found
Concavity analysis of the tangent method
The tangent method has recently been devised by Colomo and Sportiello
(arXiv:1605.01388 [math-ph]) as an efficient way to determine the shape of
arctic curves. Largely conjectural, it has been tested successfully in a
variety of models. However no proof and no general geometric insight have been
given so far, either to show its validity or to allow for an understanding of
why the method actually works. In this paper, we propose a universal framework
which accounts for the tangency part of the tangent method, whenever a
formulation in terms of directed lattice paths is available. Our analysis shows
that the key factor responsible for the tangency property is the concavity of
the entropy (also called the Lagrangean function) of long random lattice paths.
We extend the proof of the tangency to -deformed paths.Comment: published version, 22 page
Determinants of (generalised) Catalan numbers
We show that recent determinant evaluations involving Catalan numbers and
generalisations thereof have most convenient explanations by combining the
Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a
simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This
approach leads also naturally to extensions and generalisations.Comment: AmS-TeX, 16 pages; minor correction
Winding Angle Distributions for Random Walks and Flux Lines
We study analytically and numerically the winding of a flux line around a
columnar defect. Reflecting and absorbing boundary conditions apply to marginal
or repulsive defects, respectively. In both cases, the winding angle
distribution decays exponentially for large angles, with a decay constant
depending only on the boundary condition, but not on microscopic features. New
{\it non-universal} distributions are encountered for {\it chiral} defects
which preferentially twist the flux line in one direction. The resulting
asymmetric distributions have decay constants that depend on the degree of
chirality. In particular, strong chirality encourages entanglements and leads
to broad distributions. We also examine the windings of flux lines in the
presence of point impurities (random bonds). Our results suggest that pinning
to impurities reduces entanglements, leading to a narrow (Gaussian)
distribution.Comment: 12 pages Revtex and 9 postscript figure
Lattice Magnetic Walks
Sums of walks for charged particles (e.g. Hofstadter electrons) on a square
lattice in the presence of a magnetic field are evaluated. Returning loops are
systematically added to directed paths to obtain the unrestricted propagators.
Expressions are obtained for special values of the magnetic flux-per-plaquette
commensurate with the flux quantum. For commensurate and incommensurate values
of the flux, the addition of small returning loops does not affect the general
features found earlier for directed paths. Lattice Green's functions are also
obtained for staggered flux configurations encountered in models of high-Tc
superconductors.Comment: 31 pages, Plain TeX, 2 figures (available upon request),
UR-CM-93-10-1
Developing a Mathematical Model for Bobbin Lace
Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as
Computer Scientists and Mathematician
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions
We present the solution of a linear Restricted Solid--on--Solid (RSOS) model
in a field. Aside from the origins of this model in the context of describing
the phase boundary in a magnet, interest also comes from more recent work on
the steady state of non-equilibrium models of molecular motors. While similar
to a previously solved (non-restricted) SOS model in its physical behaviour,
mathematically the solution is more complex. Involving basic hypergeometric
functions , it introduces a new form of solution to the lexicon of
directed lattice path generating functions.Comment: 10 pages, 2 figure
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