860 research outputs found
Applications of Graphical Condensation for Enumerating Matchings and Tilings
A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text
revised, but content essentially the sam
From Aztec diamonds to pyramids: steep tilings
We introduce a family of domino tilings that includes tilings of the Aztec
diamond and pyramid partitions as special cases. These tilings live in a strip
of of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word and for different types of boundary
conditions, we obtain a nice product formula for the generating function of the
associated tilings with respect to the number of flips, that admits a natural
multivariate generalization. The main tools are a bijective correspondence with
sequences of interlaced partitions and the vertex operator formalism (which we
slightly extend in order to handle Littlewood-type identities). In
probabilistic terms our tilings map to Schur processes of different types
(standard, Pfaffian and periodic). We also introduce a more general model that
interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6,
new improved proof of Proposition 11
Ribbon Tilings and Multidimensional Height Functions
We fix and say a square in the two-dimensional grid indexed by
has color if . A {\it ribbon tile} of order is a
connected polyomino containing exactly one square of each color. We show that
the set of order- ribbon tilings of a simply connected region is in
one-to-one correspondence with a set of {\it height functions} from the
vertices of to satisfying certain difference restrictions.
It is also in one-to-one correspondence with the set of acyclic orientations of
a certain partially oriented graph.
Using these facts, we describe a linear (in the area of ) algorithm for
determining whether can be tiled with ribbon tiles of order and
producing such a tiling when one exists. We also resolve a conjecture of Pak by
showing that any pair of order- ribbon tilings of can be connected by a
sequence of local replacement moves. Some of our results are generalizations of
known results for order-2 ribbon tilings (a.k.a. domino tilings). We also
discuss applications of multidimensional height functions to a broader class of
polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new
references, a new illustration, and a few cosmetic changes). To appear in
Transactions of the American Mathematical Societ
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
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