arXiv subjects

Gabriel Berzunza Ojeda

Publications and source records attributed to Gabriel Berzunza Ojeda.

13 recordsLinked to original sources

arXiv2026

On the speed of coming down from infinity for subcritical branching processes with pairwise interactions

In this paper, we study the phenomenon of coming down from infinity for subcritical cooperative branching processes with pairwise interactions (BPI processes) under suitable conditions. BPI processes are continuous-time Markov chains that extend classical branching models by incorporating additional mechanisms accounting for both competitive and cooperative interactions between pairs of individuals. Our main focus is on characterising the speed at which BPI processes evolve when starting from a very large initial population in the subcritical regime. In addition, we investigate their second-order fluctuations. Furthermore, our results also apply to a class of exchangeable fragmentation-coalescent processes introduced by Berestycki (2004) and several other models from population genetics.

math.PR↗

arXiv2026

A tightness criterion for fragmentations

This note presents a simple criterion for the tightness of stochastic fragmentation processes. Our work is motivated by an application to a fragmentation process derived from deleting edges in a conditioned Galton-Watson tree studied by Berzunza-Ojeda and Holmgren (2023). In that paper, while finite-dimensional convergence was established, the claimed functional convergence relied on Lemma 22 of Broutin and Marckert (2016), which is unfortunately incorrect. We show how our results can correct the proof of Berzunza-Ojeda and Holmgren (2023). Furthermore, we show the applicability of our results by establishing tightness for fragmentation processes derived from various random tree models previously studied in the literature, such as Cayley trees, trees with specified degree sequences, and $\mathbf{p}$-trees.

math.PR↗

arXiv2026

Large fringe trees for random trees with given vertex degrees

This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case where the target tree grows with the size of the random tree. We consider three primary subtree counts: the number of fringe trees isomorphic to a specific growing tree, the number of fringe trees sharing a given growing degree statistic, and the number of fringe trees of a specific growing size. To establish our results, we employ and compare four distinct probabilistic frameworks: the method of moments with the Gao-Wormald theorem, Stein's method with coupling (to provide explicit error bounds in total variation distance), the Cai-Devroye method, and Stein's method with exchangeable pairs. Our findings provide conditions for Poisson and normal convergence for these subtree counts. Additionally, we provide a local limit theorem for sums of values obtained via sampling without replacement that may be of independent interest. Finally, our results and methods are also applied to conditioned critical Galton-Watson trees.

math.PR↗

arXiv2026

On the excision of Brownian bridge paths

Path transformations are fundamental to the study of Brownian motion and related stochastic processes, offering elegant constructions of the Brownian bridge, meander, and excursion. Central to this theory is the well-established link between Brownian motion and the $3$-dimensional Bessel process ${\rm BES}(3)$. This paper is specifically motivated by Pitman and Yor (2003), who showed that a ${\rm BES}(3)$ process can be constructed by excising the excursions of a Brownian path below its past maximum that reach zero and concatenating the remaining excursions. Our main result shows that a similar excision procedure, when applied to a Brownian bridge, can be related to a $3$-dimensional Bessel bridge.

math.PR↗

arXiv2025

Central limit theorem for supercritical Crump-Mode-Jagers processes counted with non-individual random characteristics

Consider a supercritical Crump-Mode-Jagers process $(\mathcal{Z}_{t}^φ)_{t \geq 0}$ counted with a random characteristic $φ$ that depends on an individual's life and their descendant process up to a fixed generation. Under second moment assumptions, we establish a central limit theorem for $\mathcal{Z}_{t}^φ$ as $t \rightarrow \infty$. Our result extends the recent work of Iksanov, Kolesko, and Meiners (2024) by relaxing their assumption of independent characteristics across individuals. We further demonstrate the applicability of our results to the study of fringe trees in several important random tree families, thereby providing insights into questions raised by Holmgren and Janson (2017).

math.PR↗

arXiv2025

Convergence of trees with a given degree sequence and of their associated laminations

In this paper, we study uniform rooted plane trees with given degree sequence. We show, under some natural hypotheses on the degree sequence, that these trees converge toward the so-called Inhomogeneous Continuum Random Tree after renormalisation. Our proof relies on the convergence of a modification of the well-known Lukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting-down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations that are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.

math.PR↗

arXiv2025

Convergence of the pruning processes of stable Galton-Watson trees

Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue Lévy trees. Löhr, Voisin and Winter (2015) introduced the space of bi-measure $\mathbb{R}$-trees equipped with the so-called leaf sampling weak vague topology which allows them to unify the discrete and the continuous picture by considering them as instances of the same Feller-continuous Markov process with different initial conditions. Moreover, the authors show that these so-called pruning processes converge in the Skorokhod space of càdlàg paths with values in the space of bi-measure $\mathbb{R}$-trees, whenever the initial bi-measure $\mathbb{R}$-trees converge. In this paper we provide an application to the above principle by verifying that a sequence of suitably rescaled critical conditioned Galton-Watson trees whose offspring distributions lie in the domain of attraction of a stable law of index $α\in (1,2]$ converge to the $α$-stable Lévy-tree in the leaf-sampling weak vague topology.

math.PR↗

arXiv2025

Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n \rightarrow \infty$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner. In this work, we continue the above investigation and study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $\mathbf{t}_{n}$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $α\in (1,2]$. Our main results establish that, after rescaling, the fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $α$-stable Lévy tree of index $α\in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized $α$-stable Lévy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.

math.PR↗

arXiv2025

Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees whose vertex set is a subset of $V$ which is stationary under the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with re-grafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is, with respect to the Gromov-weak topology, the scaling limit of the uniform spanning tree on $\mathbb{Z}_N^d$, $d\ge 5$. This result was recently strengthens in Archer, Nachmias and Shalev (2024) to convergence with respect to the Gromov-Hausdorff-weak topology, and therefore also with respect to the Gromov-Hausdorff topology. In the present paper we show that also the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. We give conditions on the increasing graph sequence under which the result extends to regular graphs and give probabilistic expressions scales at which time has to be speed up and edge lengths have to be scaled down.

math.PR↗

arXiv2024

Branching processes with pairwise interactions

In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on $[0,1]$ which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions.

math.PR↗

arXiv2023

Fringe trees for random trees with given vertex degrees

We prove asymptotic normality for the number of fringe subtrees isomorphic to any given tree in uniformly random trees with given vertex degrees. As applications, we also prove corresponding results for random labelled trees with given vertex degrees, for random simply generated trees (or conditioned Galton--Watson trees), and for additive functionals. The key tool for our work is an extension to the multivariate setting of a theorem by Gao and Wormald (2004), which provides a way to show asymptotic normality by analysing the behaviour of sufficiently high factorial moments.

math.PR↗

arXiv2021

The distance profile of rooted and unrooted simply generated trees

It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $α<1$, and that it is a.e. differentiable. We note that it cannot be differentiable at $0$, but leave as open questions whether it is Lipschitz, and whether is continuously differentiable on the half-line $(0,\infty)$. The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

math.PR↗

arXiv2020

The existence of a giant cluster for percolation on large Crump-Mode-Jagers trees

In this paper, we consider random trees associated with the genealogy of Crump-Mode-Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump-Mode-Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin, as well as (more general) m-ary search trees, fragmentation trees, median-of-(2l+1) binary search trees, to name a few, where up to our knowledge percolation has not been studied yet.

math.PR↗