57 research outputs found
A Reciprocity Theorem for Monomer-Dimer Coverings
The problem of counting monomer-dimer coverings of a lattice is a
longstanding problem in statistical mechanics. It has only been exactly solved
for the special case of dimer coverings in two dimensions. In earlier work,
Stanley proved a reciprocity principle governing the number of dimer
coverings of an by rectangular grid (also known as perfect matchings),
where is fixed and is allowed to vary. As reinterpreted by Propp,
Stanley's result concerns the unique way of extending to so
that the resulting bi-infinite sequence, for , satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that is always an integer satisfying the relation where unless 2(mod 4) and
is odd, in which case . Furthermore, Propp's method is
applicable to higher-dimensional cases. This paper discusses similar
investigations of the numbers , of monomer-dimer coverings, or
equivalently (not necessarily perfect) matchings of an by rectangular
grid. We show that for each fixed there is a unique way of extending
to so that the resulting bi-infinite sequence, for , satisfies a linear recurrence relation with constant coefficients. We
show that , a priori a rational number, is always an integer, using a
generalization of the combinatorial model offered by Propp. Lastly, we give a
new statement of reciprocity in terms of multivariate generating functions from
which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices
Step by step completion of a left-to-right tiling of a rectangular floor with
tiles of a single shape starts from one edge of the floor, considers the
possible ways of inserting a tile at the leftmost uncovered square, passes
through a sequence of rugged shapes of the front line between covered and
uncovered regions of the floor, and finishes with a straight front line at the
opposite edge. We count the tilings by mapping the front shapes to nodes in a
digraph, then counting closed walks on that digraph with the transfer matrix
method.
Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3
and modestly wide floors. Equivalent results are shown for the 3-dimensional
analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or
1 x 2 x 3 into rectangular containers of small cross sections.Comment: 21 pages, 21 figure
Macroscopically separated gaps in dimer coverings of Aztec rectangles
In this paper we determine the interaction of diagonal defect clusters in
regions of an Aztec rectangle that scale to arbitrary points on its symmetry
axis (in earlier work we treated the case when this point was the center of the
scaled Aztec rectangle). We use the resulting formulas to determine the
asymptotics of the correlation of defects that are macroscopically separated
from one another and feel the influence of the boundary. In several of the
treated situations this seems not to be accomplishable by previous methods. Our
applications include the case of two long neutral strings, which turn out to
interact by an analog of the Casimir force, two families of neutral doublets
that turn out to interact completely independently of one another, a neutral
doublet and a very long neutral string, a general collection of macroscopically
separated monomer and separation defects, and the case of long strings
consisting of consecutive monomers.Comment: 43 page
Short-ranged RVB physics, quantum dimer models and Ising gauge theories
Quantum dimer models are believed to capture the essential physics of
antiferromagnetic phases dominated by short-ranged valence bond configurations.
We show that these models arise as particular limits of Ising (Z_2) gauge
theories, but that in these limits the system develops a larger local U(1)
invariance that has different consequences on different lattices. Conversely,
we note that the standard Z_2 gauge theory is a generalised quantum dimer
model, in which the particular relaxation of the hardcore constraint for the
dimers breaks the U(1) down to Z_2. These mappings indicate that at least one
realization of the Senthil-Fisher proposal for fractionalization is exactly the
short ranged resonating valence bond (RVB) scenario of Anderson and of
Kivelson, Rokhsar and Sethna. They also suggest that other realizations will
require the identification of a local low energy, Ising link variable {\it and}
a natural constraint. We also discuss the notion of topological order in Z_2
gauge theories and its connection to earlier ideas in RVB theory. We note that
this notion is not central to the experiment proposed by Senthil and Fisher to
detect vortices in the conjectured Z_2 gauge field.Comment: 17 pages, 4 postscript figures automatically include
The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams
This work presents formulas for the Kauffman bracket and Jones polynomials of
3-bridge knots using the structure of Chebyshev knots and their billiard table
diagrams. In particular, these give far fewer terms than in the Skein relation
expansion. The subject is introduced by considering the easier case of 2-bridge
knots, where some geometric interpretation is provided, as well, via
combinatorial tiling problems.Comment: 20 pages, 4 figures, 2 table
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