17,571 research outputs found
A generalization of Aztec diamond theorem, part I
We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and
J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic
Combinatoric, 1992) by showing that the numbers of tilings of a certain family
of regions in the square lattice with southwest-to-northeast diagonals drawn in
are given by powers of 2. We present a proof for the generalization by using a
bijection between domino tilings and non-intersecting lattice paths.Comment: 18 page
Osculating Paths and Oscillating Tableaux
The combinatorics of certain osculating lattice paths is studied, and a
relationship with oscillating tableaux is obtained. More specifically, the
paths being considered have fixed start and end points on respectively the
lower and right boundaries of a rectangle in the square lattice, each path can
take only unit steps rightwards or upwards, and two different paths are
permitted to share lattice points, but not to cross or share lattice edges.
Such paths correspond to configurations of the six-vertex model of statistical
mechanics with appropriate boundary conditions, and they include cases which
correspond to alternating sign matrices and various subclasses thereof.
Referring to points of the rectangle through which no or two paths pass as
vacancies or osculations respectively, the case of primary interest is tuples
of paths with a fixed number of vacancies and osculations. It is then shown
that there exist natural bijections which map each such path tuple to a
pair , where is an oscillating tableau of length (i.e., a
sequence of partitions, starting with the empty partition, in which the
Young diagrams of successive partitions differ by a single square), and is
a certain, compatible sequence of weakly increasing positive integers.
Furthermore, each vacancy or osculation of corresponds to a partition in
whose Young diagram is obtained from that of its predecessor by
respectively the addition or deletion of a square. These bijections lead to
enumeration formulae for osculating paths involving sums over oscillating
tableaux.Comment: 65 pages; expanded versio
A Generalization of Aztec Diamond Theorem, Part II
The author gave a proof of a generalization of the Aztec diamond theorem for
a family of -vertex regions on the square lattice with
southwest-to-northeast diagonals drawn in (Electron. J. Combin., 2014) by using
a bijection between tilings and non-intersecting lattice paths. In this paper,
we use Kuo graphical condensation to give a new proof.Comment: 11 pages and 7 figure
Boundary Energies and the Geometry of Phase Separation in Double--Exchange Magnets
We calculate the energy of a boundary between ferro- and antiferromagnetic
regions in a phase separated double-exchange magnet in two and three
dimensions. The orientation dependence of this energy can significantly affect
the geometry of the phase-separated state in two dimensions, changing the
droplet shape and possibly stabilizing a striped arrangement within a certain
range of the model parameters. A similar effect, albeit weaker, is also present
in three dimensions. As a result, a phase-separated system near the percolation
threshold is expected to possess intrinsic hysteretic transport properties,
relevant in the context of recent experimental findings.Comment: 6 pages, including 4 figures; expanded versio
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