arXiv subjects
Publications and source records attributed to William Fleurat.
We present a simple trimming algorithm that generates nested uniform binary plane trees by removing leaves one-by-one using a best-of-three-match procedure. While its one-step transition specializes to the Luczak-Winkler & Caraceni-Stauffer coupling, its scaling limit provides a suprising c\`adl\`ag erasure of Brownian trees, reminiscent of SLE theory.
Can we obtain a Brownian CRT of mass $1/2$ from a CRT of mass $1$ by cutting certain branches? In this paper, we will answer that question in the much more general setting of self-similar Markov trees. Self-similar Markov trees (ssMt) are random decorated trees that encode the genealogy of a system of particles carrying positive labels, and where particles undergo splitting and growth depending on their labels in a self-similar fashion. Introduced and developed in the recent monograph (Bertoin-Curien-Riera, 2024), they provide a broad generalization of Brownian and stable continuum random trees and arise naturally in various models of random geometry such as the Brownian sphere/disk. The law of a ssMt is characterized by its quadruplet $(\mathrm{a}, \sigma^2, \boldsymbol{\Lambda}; \alpha)$, which specifies the features of the underlying growth-fragmentation mechanism, together with the initial decoration $x>0$. In this work, we focus on special cases of ssMt in which the trees started from different initial values $x>0$ can be coupled into a continuous, increasing family of nested subtrees. In the case of the Brownian and stable continuum random trees, this yields surprisingly simple novel dynamics corresponding to the scaling limit of the leaf-growth algorithms of Luczak-Winkler and Caraceni-Stauffer.
We prove sandwich theorems and a Tauberian theorem in the space of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prokhorov (GHP) topology. These results hold with respect to a close relative of Gromov's Lipschitz order. As a proof-of-concept of a general method to prove metric scaling limits of random discrete structures, we give an application to the theory of random planar maps: the Brownian sphere is the scaling limit in the GHP topology of irreducible quandrangulations. Our main inputs are (i) the convergence of general quadrangulations to the Brownian sphere (Le Gall, 2013; Miermont, 2013); and (ii) couplings where irreducible quadrangulations of the hexagon are "grown" by face-openings (Addario-Berry, 2014).
We show that given a log-concave offspring distribution, the corresponding sequence of Bienaym\'e-Galton-Watson trees conditioned to have $n\geq 1$ vertices admits a realization as a Markov process $(T_n)_{n\geq1}$ which adds a new "right-leaning" leaf at each step. This applies for instance to offspring distributions which are Poisson, binomial, geometric, or any convolution of those. By a negative result of Janson, the log-concavity condition is optimal in the restricted case of offspring distributions supported in $\{0,1,2\}$. We then prove a generalization to the case of an offspring distribution supported on an arithmetic progression, if we assume log-concavity along that progression. As an application, we deduce the existence of increasing couplings in an inhomogeneous model of random subtrees of the Ulam--Harris tree. This is equivalent to the statement that, in a corresponding inhomogeneous Bernouilli percolation model on a regular tree, the root cluster is stochastically increasing in its size. These results generalize a construction of Luczak and Winkler which applies to uniformly sampled subtrees with $n$ vertices of the infinite complete $d$-ary trees. Our proofs are elementary and we tried to make them as self-contained as possible.
We consider the model of random planar maps of size $n$ biased by a weight $u>0$ per $2$-connected block, and the closely related model of random planar quadrangulations of size $n$ biased by a weight $u>0$ per simple component. We exhibit a phase transition at the critical value $u_C=9/5$. If $u u_C$, the largest block is of size $Θ(\log(n))$, the scaling order for distances is $n^{1/2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $Θ(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.