2,948 research outputs found
On Brownian limits of planar trees and maps with a prescribed degree sequence
We study a configuration model on bipartite planar maps where, given even
integers, one samples a planar map uniformly at random with these face degrees.
We prove that when suitably rescaled, such maps always admit subsequential
limits as in the Gromov-Hausdorff-Prokhorov topology. Further,
we show that they converge in distribution towards the celebrated Brownian map,
and more generally a Brownian disk for maps with a boundary, if and only if
there is no inner face with a macroscopic degree, or, if the perimeter is too
big, the maps degenerate and converge to the Brownian CRT. The latter case
include that of size-conditioned Boltzmann map associated with critical weights
in the domain of attraction of a Cauchy distribution, which was missing in the
literature. Our proofs rely on bijections with random labelled plane trees,
which are similarly sampled uniformly given outdegrees. Along the way, we
obtain some results on the geometry of such trees, such as a convergence to the
Brownian CRT but only in the weaker sense of subtrees spanned by random
vertices, which are of independent interest.Comment: The previous version has been merged with arXiv:1902.0453
Fires on large recursive trees
We consider random dynamics on a uniform random recursive tree with
vertices. Successively, in a uniform random order, each edge is either set on
fire with some probability or fireproof with probability . Fires
propagate in the tree and are only stopped by fireproof edges. We first
consider the proportion of burnt and fireproof vertices as , and
prove a phase transition when is of order . We then study the
connectivity of the fireproof forest, more precisely the existence of a giant
component. We finally investigate the sizes of the burnt subtrees.Comment: Accepted for publication in Stochastic Processes and their
Applications. 24 pages, 4 figure
Triangulating stable laminations
We study the asymptotic behavior of random simply generated noncrossing
planar trees in the space of compact subsets of the unit disk, equipped with
the Hausdorff distance. Their distributional limits are obtained by
triangulating at random the faces of stable laminations, which are random
compact subsets of the unit disk made of non-intersecting chords coded by
stable L\'evy processes. We also study other ways to "fill-in" the faces of
stable laminations, which leads us to introduce the iteration of laminations
and of trees.Comment: 34 pages, 5 figure
Exploiting network topology for large-scale inference of nonlinear reaction models
The development of chemical reaction models aids understanding and prediction
in areas ranging from biology to electrochemistry and combustion. A systematic
approach to building reaction network models uses observational data not only
to estimate unknown parameters, but also to learn model structure. Bayesian
inference provides a natural approach to this data-driven construction of
models. Yet traditional Bayesian model inference methodologies that numerically
evaluate the evidence for each model are often infeasible for nonlinear
reaction network inference, as the number of plausible models can be
combinatorially large. Alternative approaches based on model-space sampling can
enable large-scale network inference, but their realization presents many
challenges. In this paper, we present new computational methods that make
large-scale nonlinear network inference tractable. First, we exploit the
topology of networks describing potential interactions among chemical species
to design improved "between-model" proposals for reversible-jump Markov chain
Monte Carlo. Second, we introduce a sensitivity-based determination of move
types which, when combined with network-aware proposals, yields significant
additional gains in sampling performance. These algorithms are demonstrated on
inference problems drawn from systems biology, with nonlinear differential
equation models of species interactions
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