16,242 research outputs found
Representations of stack triangulations in the plane
Stack triangulations appear as natural objects when defining an increasing
family of triangulations by successive additions of vertices. We consider two
different probability distributions for such objects. We represent, or "draw"
these random stack triangulations in the plane and study the asymptotic
properties of these drawings, viewed as random compact metric spaces. We also
look at the occupation measure of the vertices, and show that for these two
distributions it converges to some random limit measure.Comment: 29 pages, 13 figure
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
Calibrated Tree Priors for Relaxed Phylogenetics and Divergence Time Estimation
The use of fossil evidence to calibrate divergence time estimation has a long
history. More recently Bayesian MCMC has become the dominant method of
divergence time estimation and fossil evidence has been re-interpreted as the
specification of prior distributions on the divergence times of calibration
nodes. These so-called "soft calibrations" have become widely used but the
statistical properties of calibrated tree priors in a Bayesian setting has not
been carefully investigated. Here we clarify that calibration densities, such
as those defined in BEAST 1.5, do not represent the marginal prior distribution
of the calibration node. We illustrate this with a number of analytical results
on small trees. We also describe an alternative construction for a calibrated
Yule prior on trees that allows direct specification of the marginal prior
distribution of the calibrated divergence time, with or without the restriction
of monophyly. This method requires the computation of the Yule prior
conditional on the height of the divergence being calibrated. Unfortunately, a
practical solution for multiple calibrations remains elusive. Our results
suggest that direct estimation of the prior induced by specifying multiple
calibration densities should be a prerequisite of any divergence time dating
analysis
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
Inferring clonal evolution of tumors from single nucleotide somatic mutations
High-throughput sequencing allows the detection and quantification of
frequencies of somatic single nucleotide variants (SNV) in heterogeneous tumor
cell populations. In some cases, the evolutionary history and population
frequency of the subclonal lineages of tumor cells present in the sample can be
reconstructed from these SNV frequency measurements. However, automated methods
to do this reconstruction are not available and the conditions under which
reconstruction is possible have not been described.
We describe the conditions under which the evolutionary history can be
uniquely reconstructed from SNV frequencies from single or multiple samples
from the tumor population and we introduce a new statistical model, PhyloSub,
that infers the phylogeny and genotype of the major subclonal lineages
represented in the population of cancer cells. It uses a Bayesian nonparametric
prior over trees that groups SNVs into major subclonal lineages and
automatically estimates the number of lineages and their ancestry. We sample
from the joint posterior distribution over trees to identify evolutionary
histories and cell population frequencies that have the highest probability of
generating the observed SNV frequency data. When multiple phylogenies are
consistent with a given set of SNV frequencies, PhyloSub represents the
uncertainty in the tumor phylogeny using a partial order plot. Experiments on a
simulated dataset and two real datasets comprising tumor samples from acute
myeloid leukemia and chronic lymphocytic leukemia patients demonstrate that
PhyloSub can infer both linear (or chain) and branching lineages and its
inferences are in good agreement with ground truth, where it is available
Some families of increasing planar maps
Stack-triangulations appear as natural objects when one wants to define some
increasing families of triangulations by successive additions of faces. We
investigate the asymptotic behavior of rooted stack-triangulations with
faces under two different distributions. We show that the uniform distribution
on this set of maps converges, for a topology of local convergence, to a
distribution on the set of infinite maps. In the other hand, we show that
rescaled by , they converge for the Gromov-Hausdorff topology on
metric spaces to the continuum random tree introduced by Aldous. Under a
distribution induced by a natural random construction, the distance between
random points rescaled by converge to 1 in probability.
We obtain similar asymptotic results for a family of increasing
quadrangulations
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