On scaling limits of multitype Galton-Watson trees with possibly infinite variance

In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive L\'evy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offspring distribution. Our approach relies on a remarkable decomposition for multitype trees into monotype trees introduced by Miermont.Comment: 30 pages, 2 figure

The fluctuations of the giant cluster for percolation on random split trees

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.Comment: 43 page

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 $\alpha \in (1,2]$. 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 $\alpha$-stable L\'evy tree of index $\alpha \in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\'evy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.Comment: 30 pages, 5 figure

Branching processes with pairwise interactions

In this manuscript, we are interested on 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 individual are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by Gonz\'alez Casanova et al. (2017), and includes the so-called logistic branching process which was studied by Lambert (2005). Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments uses the moment dual of BPI-processes which turns out to be a family of diffusions taking values on $[0,1]$, that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries.Comment: 3 table

The k-Cut Model in Conditioned Galton-Watson Trees

The k-cut number of rooted graphs was introduced by Cai et al. [Cai and Holmgren, 2019] as a generalization of the classical cutting model by Meir and Moon [Meir and Moon, 1970]. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson [Janson, 2006]

The kk-cut model in deterministic and random trees

The $k$-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the k-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees

Leishmania (L.) mexicana infected bats in Mexico: novel potential reservoirs

Leishmania (Leishmania) mexicana causes cutaneous leishmaniasis, an endemic zoonosis affecting a growing number of patients in the southeastern states of Mexico. Some foci are found in shade-grown cocoa and coffee plantations, or near perennial forests that provide rich breeding grounds for the sand fly vectors, but also harbor a variety of bat species that live off the abundant fruits provided by these shade-giving trees. The close proximity between sand flies and bats makes their interaction feasible, yet bats infected with Leishmania (L.) mexicana have not been reported. Here we analyzed 420 bats from six states of Mexico that had reported patients with leishmaniasis. Tissues of bats, including skin, heart, liver and/or spleen were screened by PCR for Leishmania (L.) mexicana DNA. We found that 41 bats (9.77%), belonging to 13 species, showed positive PCR results in various tissues. The infected tissues showed no evidence of macroscopic lesions. Of the infected bats, 12 species were frugivorous, insectivorous or nectarivorous, and only one species was sanguivorous (Desmodus rotundus), and most of them belonged to the family Phyllostomidae. The eco-region where most of the infected bats were caught is the Gulf Coastal Plain of Chiapas and Tabasco. Through experimental infections of two Tadarida brasiliensis bats in captivity, we show that this species can harbor viable, infective Leishmania (L.) mexicana parasites that are capable of infecting BALB/c mice. We conclude that various species of bats belonging to the family Phyllostomidae are possible reservoir hosts for Leishmania (L.) mexicana, if it can be shown that such bats are infective for the sand fly vector. Further studies are needed to determine how these bats become infected, how long the parasite remains viable inside these potential hosts and whether they are infective to sand flies to fully evaluate their impact on disease epidemiology

Yule processes with rare mutation and their applications to percolation on b-ary trees

We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees