On Brownian limits of planar trees and maps with a prescribed degree sequence

We study a configuration model on bipartite planar maps where, given $n$ 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 $n \to \infty$ 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 $n$ 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 $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as $n\to\infty$, and prove a phase transition when $p_n$ is of order $\ln n/n$. 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