26 research outputs found
Scaling limit of the invasion percolation cluster on a regular tree
We prove existence of the scaling limit of the invasion percolation cluster
(IPC) on a regular tree. The limit is a random real tree with a single end. The
contour and height functions of the limit are described as certain diffusive
stochastic processes. This convergence allows us to recover and make precise
certain asymptotic results for the IPC. In particular, we relate the limit of
the rescaled level sets of the IPC to the local time of the scaled height
function.Comment: Published in at http://dx.doi.org/10.1214/11-AOP731 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A decorated tree approach to random permutations in substitution-closed classes
We establish a novel bijective encoding that represents permutations as
forests of decorated (or enriched) trees. This allows us to prove local
convergence of uniform random permutations from substitution-closed classes
satisfying a criticality constraint. It also enables us to reprove and
strengthen permuton limits for these classes in a new way, that uses a
semi-local version of Aldous' skeleton decomposition for size-constrained
Galton--Watson trees.Comment: New version including referee's corrections, accepted for publication
in Electronic Journal of Probabilit
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
Local weak convergence and propagation of ergodicity for sparse networks of interacting processes
We study the limiting behavior of interacting particle systems indexed by
large sparse graphs, which evolve either according to a discrete time Markov
chain or a diffusion, in which particles interact directly only with their
nearest neighbors in the graph. To encode sparsity we work in the framework of
local weak convergence of marked (random) graphs. We show that the joint law of
the particle system varies continuously with respect to local weak convergence
of the underlying graph. In addition, we show that the global empirical measure
converges to a non-random limit, whereas for a large class of graph sequences
including sparse Erd\"{o}s-R\'{e}nyi graphs and configuration models, the
empirical measure of the connected component of a uniformly random vertex
converges to a random limit. Finally, on a lattice (or more generally an
amenable Cayley graph), we show that if the initial configuration of the
particle system is a stationary ergodic random field, then so is the
configuration of particle trajectories up to any fixed time, a phenomenon we
refer to as "propagation of ergodicity". Along the way, we develop some general
results on local weak convergence of Gibbs measures in the uniqueness regime
which appear to be new.Comment: 46 pages, 1 figure. This version of the paper significantly extends
the convergence results for diffusions in v1, and includes new results on
propagation of ergodicity and discrete-time models. The complementary results
in v1 on autonomous characterization of marginal dynamics of diffusions on
trees and generalizations thereof are now presented in a separate paper
arXiv:2009.1166
A branching process with coalescence to model random phylogenetic networks
We introduce a biologically natural, mathematically tractable model of random
phylogenetic network to describe evolution in the presence of hybridization.
One of the features of this model is that the hybridization rate of the
lineages correlates negatively with their phylogenetic distance. We give
formulas / characterizations for quantities of biological interest that make
them straightforward to compute in practice. We show that the appropriately
rescaled network, seen as a metric space, converges to the Brownian continuum
random tree, and that the uniformly rooted network has a local weak limit,
which we describe explicitly