5,363 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
Terrestrial Planet Formation I. The Transition from Oligarchic Growth to Chaotic Growth
We use a hybrid, multiannulus, n-body-coagulation code to investigate the
growth of km-sized planetesimals at 0.4-2 AU around a solar-type star. After a
short runaway growth phase, protoplanets with masses of roughly 10^26 g and
larger form throughout the grid. When (i) the mass in these `oligarchs' is
roughly comparable to the mass in planetesimals and (ii) the surface density in
oligarchs exceeds 2-3 g/sq cm at 1 AU, strong dynamical interactions among
oligarchs produce a high merger rate which leads to the formation of several
terrestrial planets. In disks with lower surface density, milder interactions
produce several lower mass planets. In all disks, the planet formation
timescale is roughly 10-100 Myr, similar to estimates derived from the
cratering record and radiometric data.Comment: Astronomical Journal, accepted; 22 pages + 15 figures in ps format;
eps figures at http://cfa-www.harvard.edu/~kenyon/dl/ revised version
clarifies evolution and justifies choice of promotion masse
Thermonuclear runaways in thick hydrogen rich envelopes of neutron stars
A Lagrangian, fully implicit, one dimensional hydrodynamic computer code was used to evolve thermonuclear runaways in the accreted hydrogen rich envelopes of 1.0 Msub solar neutron stars with radii of 10 km and 20 km. Simulations produce outbursts which last from about 750 seconds to about one week. Peak effective temeratures and luninosities were 26 million K and 80 thousand Lsub solar for the 10 km study and 5.3 millison and 600 Lsub solar for the 20 km study. Hydrodynamic expansion on the 10 km neutron star produced a precursor lasting about one ten thousandth seconds
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