We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large functional values always converge locally to immortal trees. We also derive a very general ratio limit property for functionals of random trees satisfying the monotonicity property. Then we move on to study the local convergence of critical continuous-state branching processes, and prove a similar result. Finally we give a definition of continuum condensation trees, which should be the correct local limits for certain subcritical Levy trees under suitable conditionings.
The goal of these lectures is to survey some of the recent progress on the description of large-scale structure of random trees. We use the framework of Markov-Branching sequences of trees and discuss several applications.
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit of finite variance Galton-Watson trees condition on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to this is a generalization of the classical Otter-Dwass formula.
We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour.
By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on Lévy trees. Then we could construct a decreasing Lévy-CRT-valued process $\{{\mathcal T}_t\}$ by pruning Lévy trees and an analogous process $\{{\mathcal T}^*_t\}$ by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of $\{{\mathcal T}_t\}$ at the ascension time can be represented by $\{{\mathcal T}^*_t\}$. The results generalize those studied in Abraham and Delmas (2012).
The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. co…
This paper considers linear functions constructed on two different weighted branching processes and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich-Rubins…
The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductan…
We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that "macroscopic" splitting events are rare, we show that Markov branching trees admit the…
We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li, we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establ…
We define a multi-type coalescent point process of a general branching process with finitely many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population, providing types along ancestral lineages of all individuals in the standing population. We show that this point-process is a functional of a certain Markov chain defined by the planar embedding of the branching process. This Markov chain is further used to determine statistical properties of the ancestral tree, such as the time to the most recent common ancestor (MRCA) of two consecutive …
We consider a model of stationary population with random size given by a continuous state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure of the genealogical tree of $n$ individuals randomly chosen among the extant population at a given time. Then, we prove the convergence of the renormalized total length of this genealogical tree as $n$ goes to infinity, see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant size population. The limit appears already in Bi and Delmas (2016) but with a diff…
We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingma coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage…
We are interested in the genealogical structure of alleles for a Bienaymé-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jirina process) in discrete time. Itô's excursion theory and the Léevy-Itô decomposition of subordinators provide fundamental insights for the results.
Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks, since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree $m\geq 2$. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is…
Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz, Röllin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to…
This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees. We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types o…
We consider random recursive trees that are grown via community modulated schemes that involve random attachment or degree based attachment. The aim of this paper is to derive general techniques based on continuous time embedding to study such models. The associated continuous time embeddings are not branching processes: individual reproductive rates at each time depend on the composition of the entire population at that time. Using stochastic analytic techniques we show that various key macroscopic statistics of the continuous time embedding stabilize, allowing asymptotics for a host of funct…
Let $T$ be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on $T$, in which at each time step every particle gives birth to a random number of children with mean $d$ and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex $0$ and conditionally on survival of the process, the time it takes for every vertex within distance $r$ of $0$ to be hit by a particle of the branching random walk is almost surely $r + \frac{2}{\log(3/2)}\log\…
We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural a…
A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means M_n grow fast enough so that sum M_n^{-1} is finite. In fact, such a process is constructed for every offspring distribution of infinite variance, and this establishes the converse of a previously known fact: that if a distribution has finite variance then sum M_n^{-1}=infty is equivalent to almost sure extinction. This has as an immediate consequence the converse to a theorem on equipolarity of Galton-Watson trees.
We study a genealogical model for continuous-state branching processes with immigration with a (sub)critical branching mechanism. This model allows the immigrants to be on the same line of descent. The corresponding family tree is an ordered rooted continuum random tree with a single infinite end defined thanks to two continuous processes denoted by $(\overleftarrow{H}_t ;t\geq 0)$ and $(\overrightarrow{H}_t ;t\geq 0)$ that code the parts at resp. the left and the right hand of the infinite line of descent of the tree. These processes are called the left and the right height processes. We defi…
We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [9]. Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [17]. Surprisingly, this phase transition phenomenon disappears in the sense …
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $λ$. There is a threshold for $λ$, which is called $λ_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $λ_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $λ_s$ is completely understood and can be computed as a function of the reproduction rates. On the oth…
The Horton and Tokunaga branching laws provide a convenient framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching. The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, and proven for two paradigmatic models: the critical Galton-Watson branching process with finite progeny …
We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To…
We investigate the genealogical structure of general critical or subcritical continuous-state branching processes. Analogously to the coding of a discrete tree by its contour function, this genealogical structure is coded by a real-valued stochastic process called the height process, which is itself constructed as a local time functional of a Levy process with no negative jumps. We present a detailed study of the height process and of an associated measure-valued process called the exploration process, which plays a key role in most applications. Under suitable assumptions, we prove that whene…
Let (Zn) be a branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on logZn by using the convergence rates of the logarithm of submartingale and the result of the corresponding random walk on the Berry-Esseen bound.
In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processe…
Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.
We consider a supercritical branching process $(Z_n)$ in a random environment $ξ$. Let $W$ be the limit of the normalized population size $W_n=Z_n/E[Z_n|ξ]$. We first show a necessary and sufficient condition for the quenched $L^p$ ($p>1$) convergence of $(W_n)$, which completes the known result for the annealed $L^p$ convergence. We then show that the convergence rate is exponential, and we find the maximal value of $ρ>1$ such that $ρ^n(W-W_n)\rightarrow 0$ in $L^p$, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment.
A decomposable strongly critical Galton-Watson branching process with $N$ types of particles labelled $1,2,...,N$ is considered in which a type~$i$ parent may produce individuals of types $j\geq i$ only. This model may be viewed as a stochastic model for the sizes of a geographically structured population occupying $N$ islands, the location of a particle being considered as its type. The newborn particles of island $i\leq N-1$ either stay at the same island or migrate, just after their birth to the islands $% i+1,i+2,...,N$. Particles of island $N$ do not migrate. We investigate the structure …
A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet $(σ^2,a,Λ)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(σ^2,a,Λ)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of…
Considering a critical branching random walk on the real line. In a recent paper, Aidekon [3] developed a powerful method to obtain the convergence in law of its minimum after a log-factor normalization. By an adaptation of this method, we show that the point process formed by the branching random walk and its minimum converge in law to a Poisson point process colored by a certain point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et a…
Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $λ$ from an open set $Λ\subseteq \mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\partial Λ$ of $Λ$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the rema…
We study the distribution of the maximal jump of continuous-state branching processes. Several exact expressions and explicit asymptotics of both the local maximal jump and the global maximal jump are obtained. We also compare the distribution of the maximal jump and the Lévy measure to get several absolute continuity results. Then we study local convergence of continuous-state branching processes under various conditionings. We obtain complete results under the conditioning of large maximal jump, and partial results under two other conditionings, which are, the conditioning of large width, an…
We are interested in predator-prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase-escape process, introduced by Kordzakhia with a variant by Bordenave, and the birth-and-assassination process, introduced by Aldous & Krebs. We exhibit a coupling between these processes and branching random walks starting from a random point and killed at the barrier 0. This sheds new light on the chase-escape and birth-and-a…
Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $ξ_e$ indexed by all the edges in the tree. We assign the resistance $m^d ξ_e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.
We consider a model of random tree growth, where at each time unit a new vertex is added and attached to an already existing vertex chosen at random. The probability with which a vertex with degree $k$ is chosen is proportional to $w(k)$, where the weight function $w$ is the parameter of the model. In the papers of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and, independently, Mori, the asymptotic degree distribution is obtained for a model that is equivalent to the special case of ours, when the weight function is linear. The proof therein strongly relies on the linear choice of $w$. We…
We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$ and $\tilde Z_n(t)=\int e^{tx}Z_n(dx)$ be its Laplace transform. We show the $L^p$ convergence rate and the uniform convergence of the martingale $\tilde Z_n(t)/\mathbb E[\tilde Z_n(t)|ξ]$, and establish a moderate deviation principle for the measures $Z_n$.
We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is a least the sum of the sizes of its children. Assuming that macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased chil…
In this work, we study asymptotics of the genealogy of Galton-Watson processes. Thus we consider a offspring distribution such that the rescaled Galton-Watson processes converges to a continuous state branching process (CSBP) with jumps. After we show that the rescaled height (or exploration) process of the corresponding Galton-Watson family tree, converges in a functional sense, to the continuous height process that Le Gall and Le Jan introduced in 1998 on their paper "branching processes in Lévy processes : The exploration process".
In this paper, we show that a Galton-Watson tree conditioned to have a fixed number of particles in generation $n$ converges in distribution as $n\rightarrow\infty$, and with this tool we study the span and gap statistics of a branching random walk on such trees, which is the discrete version of Ramola, Majumdar and Schehr 2015, generalized to arbitrary offspring and displacement distributions with moment constraints.
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our tech…
We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman's martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the …
