165 research outputs found
Rayleigh processes, real trees, and root growth with re-grafting
The real trees form a class of metric spaces that extends the class of trees
with edge lengths by allowing behavior such as infinite total edge length and
vertices with infinite branching degree. Aldous's Brownian continuum random
tree, the random tree-like object naturally associated with a standard Brownian
excursion, may be thought of as a random compact real tree. The continuum
random tree is a scaling limit as N tends to infinity of both a critical
Galton-Watson tree conditioned to have total population size N as well as a
uniform random rooted combinatorial tree with N vertices. The Aldous--Broder
algorithm is a Markov chain on the space of rooted combinatorial trees with N
vertices that has the uniform tree as its stationary distribution. We construct
and study a Markov process on the space of all rooted compact real trees that
has the continuum random tree as its stationary distribution and arises as the
scaling limit as N tends to infinity of the Aldous--Broder chain. A key
technical ingredient in this work is the use of a pointed Gromov--Hausdorff
distance to metrize the space of rooted compact real trees.Comment: 48 Pages. Minor revision of version of Feb 2004. To appear in
Probability Theory and Related Field
Subtree prune and regraft: a reversible real tree-valued Markov process
We use Dirichlet form methods to construct and analyze a reversible Markov
process, the stationary distribution of which is the Brownian continuum random
tree. This process is inspired by the subtree prune and regraft (SPR) Markov
chains that appear in phylogenetic analysis. A key technical ingredient in this
work is the use of a novel Gromov--Hausdorff type distance to metrize the space
whose elements are compact real trees equipped with a probability measure.
Also, the investigation of the Dirichlet form hinges on a new path
decomposition of the Brownian excursion.Comment: Published at http://dx.doi.org/10.1214/009117906000000034 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Genealogy of catalytic branching models
We consider catalytic branching populations. They consist of a catalyst
population evolving according to a critical binary branching process in
continuous time with a constant branching rate and a reactant population with a
branching rate proportional to the number of catalyst individuals alive. The
reactant forms a process in random medium. We describe asymptotically the
genealogy of catalytic branching populations coded as the induced forest of
-trees using the many individuals--rapid branching continuum limit.
The limiting continuum genealogical forests are then studied in detail from
both the quenched and annealed points of view. The result is obtained by
constructing a contour process and analyzing the appropriately rescaled version
and its limit. The genealogy of the limiting forest is described by a point
process. We compare geometric properties and statistics of the reactant limit
forest with those of the "classical" forest.Comment: Published in at http://dx.doi.org/10.1214/08-AAP574 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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