293 research outputs found
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
Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
We show how to compute the probabilities of various connection topologies for
uniformly random spanning trees on graphs embedded in surfaces. As an
application, we show how to compute the "intensity" of the loop-erased random
walk in , that is, the probability that the walk from (0,0) to
infinity passes through a given vertex or edge. For example, the probability
that it passes through (1,0) is 5/16; this confirms a conjecture from 1994
about the stationary sandpile density on . We do the analogous
computation for the triangular lattice, honeycomb lattice and , for which the probabilities are 5/18, 13/36, and
respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised
section on the LERW intensity, and an expanded appendix on the annular matri
On the asymptotics of dimers on tori
We study asymptotics of the dimer model on large toric graphs. Let be a weighted -periodic planar graph, and let
be a large-index sublattice of . For bipartite we
show that the dimer partition function on the quotient
has the asymptotic expansion , where is the area of ,
is the free energy density in the bulk, and is a finite-size
correction term depending only on the conformal shape of the domain together
with some parity-type information. Assuming a conjectural condition on the zero
locus of the dimer characteristic polynomial, we show that an analogous
expansion holds for non-bipartite. The functional form of the
finite-size correction differs between the two classes, but is universal within
each class. Our calculations yield new information concerning the distribution
of the number of loops winding around the torus in the associated double-dimer
models.Comment: 48 pages, 18 figure
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
Psychological Factors Relevant to the Prehospital and In-hospital Phases of Acute Myocardial Infarction
Recognition and treatment of psychological factors relevant to the acute prehospital and in-hospital phases of myocardial infarction (Ml) are reviewed. Various emotions and personality characteristics can be both risk factors for and consequences of acute Ml. Components of the Type A behavior pattern and levels of somatic and emotional awareness have been linked with excessive treatment-seeking delay for Ml patients. Psychiatric conditions such as panic disorder may mimic symptomatic presentation of Ml and therefore have implications for differential diagnosis in the emergency room. Additionally. anxiety, depression, and neurobehavioral disorders such as delirium are relatively common during the hospitalization period and may contribute to potentially lethal complications of Ml. Because psychological factors are associated with prognosis during each phase of Ml, the identification and treatment of such factors are crucial in providing comprehensive care for MI patients
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