347 research outputs found
Arctic curves of the octahedron equation
We study the octahedron relation (also known as the -system),
obeyed in particular by the partition function for dimer coverings of the Aztec
Diamond graph. For a suitable class of doubly periodic initial conditions, we
find exact solutions with a particularly simple factorized form. For these, we
show that the density function that measures the average dimer occupation of a
face of the Aztec graph, obeys a system of linear recursion relations with
periodic coefficients. This allows us to explore the thermodynamic limit of the
corresponding dimer models and to derive exact "arctic" curves separating the
various phases of the system.Comment: 39 pages, 21 figures; typos fixed, four references and an appendix
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Arctic circles, domino tilings and square Young tableaux
The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly
random domino tilings of an Aztec diamond of high order are frozen with
asymptotically high probability outside the "arctic circle" inscribed within
the diamond. A similar arctic circle phenomenon has been observed in the
limiting behavior of random square Young tableaux. In this paper, we show that
random domino tilings of the Aztec diamond are asymptotically related to random
square Young tableaux in a more refined sense that looks also at the behavior
inside the arctic circle. This is done by giving a new derivation of the
limiting shape of the height function of a random domino tiling of the Aztec
diamond that uses the large-deviation techniques developed for the square Young
tableaux problem in a previous paper by Pittel and the author. The solution of
the variational problem that arises for domino tilings is almost identical to
the solution for the case of square Young tableaux by Pittel and the author.
The analytic techniques used to solve the variational problem provide a
systematic, guess-free approach for solving problems of this type which have
appeared in a number of related combinatorial probability models.Comment: Published in at http://dx.doi.org/10.1214/10-AOP628 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Boundary Partitions in Trees and Dimers
Given a finite planar graph, a grove is a spanning forest in which every
component tree contains one or more of a specified set of vertices (called
nodes) on the outer face. For the uniform measure on groves, we compute the
probabilities of the different possible node connections in a grove. These
probabilities only depend on boundary measurements of the graph and not on the
actual graph structure, i.e., the probabilities can be expressed as functions
of the pairwise electrical resistances between the nodes, or equivalently, as
functions of the Dirichlet-to-Neumann operator (or response matrix) on the
nodes. These formulae can be likened to generalizations (for spanning forests)
of Cardy's percolation crossing probabilities, and generalize Kirchhoff's
formula for the electrical resistance. Remarkably, when appropriately
normalized, the connection probabilities are in fact integer-coefficient
polynomials in the matrix entries, where the coefficients have a natural
algebraic interpretation and can be computed combinatorially. A similar
phenomenon holds in the so-called double-dimer model: connection probabilities
of boundary nodes are polynomial functions of certain boundary measurements,
and as formal polynomials, they are specializations of the grove polynomials.
Upon taking scaling limits, we show that the double-dimer connection
probabilities coincide with those of the contour lines in the Gaussian free
field with certain natural boundary conditions. These results have direct
application to connection probabilities for multiple-strand SLE_2, SLE_8, and
SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor
change
Asymptotics of multivariate sequences, part III: quadratic points
We consider a number of combinatorial problems in which rational generating
functions may be obtained, whose denominators have factors with certain
singularities. Specifically, there exist points near which one of the factors
is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the
coefficients of such a generating function. The computation requires some
topological deformations as well as Fourier-Laplace transforms of generalized
functions. We apply the results of the theory to specific combinatorial
problems, such as Aztec diamond tilings, cube groves, and multi-set
permutations.Comment: substantial correction
Domino statistics of the two-periodic Aztec diamond
Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary
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