Branching Random Walk Research Articles

Harmonic Analysis of Branching Random Walks with Heavy Tails

We consider a continuous-time symmetric, spatially homogeneous branching random walk on a multidimensional lattice with a single branching source. Corresponding transition intensities of the underlying random walk are assumed to have heavy tails. This assumption implies that the variance of jumps is infinite. The growth rate estimate of the Fourier transform for transition intensities and of the asymptotics of the mean number of particles in the source in subcritical case are obtained.

On large-deviation probabilities for the empirical distribution of branching random walks with heavy tails

AbstractGiven a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).

Branching Random Walks with Two Types of Particles on Multidimensional Lattices

We consider a continuous-time branching random walk on a multidimensional lattice with two types of particles and an infinite number of initial particles. The main results are devoted to the study of the generating function and the limiting behavior of the moments of subpopulations generated by a single particle of each type. We assume that particle types differ from each other not only by the laws of branching, as in multi-type branching processes, but also by the laws of walking. For a critical branching process at each lattice point and recurrent random walk of particles, the effect of limit spatial clustering of particles over the lattice is studied. A model illustrating epidemic propagation is also considered. In this model, we consider two types of particles: infected and immunity generated. Initially, there is an infected particle that can infect others. Here, for the local number of particles of each type at a lattice point, we study the moments and their limiting behavior. Additionally, the effect of intermittency of the infected particles is studied for a supercritical branching process at each lattice point. Simulations are presented to demonstrate the effect of limit clustering for the epidemiological model.

Capacity of the Range of Branching Random Walks in Low Dimensions

Consider a branching random walk $(V_u)_{u\in \mathcal T^{IGW}}$ in $\mathbb Z^d$ with the genealogy tree $\mathcal T^{IGW}$ formed by a sequence of i.i.d. critical Galton-Watson trees. Let $R_n $ be the set of points in $\mathbb Z^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in $\mathcal T^{IGW}$. Our main result states that for $d\in \{3, 4, 5\}$, the capacity of $R_n$ is almost surely equal to $n^{\frac{d-2}{2}+o(1)}$ as $n \to \infty$.

The Probability of Reaching a Receding Boundary by a Branching Random Walk with Fading Branching and Heavy-Tailed Jump Distribution

Фосс и Захари (2003) и Фосс, Пальмовски и Захари (2005) изучали вероятность достижения удаляющейся границы на интервале времени случайной длины случайным блужданием, распределение скачков которого имеет тяжелый хвост. Они предложили и развили новый подход, который позволил обобщить результаты Асмуссена (1998) на случай произвольных моментов остановки и широкого класса нелинейных границ и получить равномерные утверждения по всем моментам остановки. В данной работе рассмотрен один класс ветвящихся случайных блужданий с затуханием ветвления, для которого получены утверждения об асимптотике максимума значений ветвящегося случайного блуждания на интервале времени случайной (возможно, неограниченной) длины, а также равномерные утверждения по классу ограниченных случайных интервалов времени.

The Probability of Reaching a Receding Boundary by a Branching Random Walk with Fading Branching and Heavy-Tailed Jump Distribution

Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and developed a new approach that allows to generalise results of Asmussen (1998) onto the case of arbitrary stopping times and a wide class of nonlinear boundaries, and to obtain uniform results over all stopping times. In this paper, we consider a class of branching random walks with fading branching and obtain results on the tail asymptotics for the maximum of a branching random walk on a time interval of random (possibly unlimited) length, as well as uniform results within a class of bounded random time intervals.

Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment

Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment

Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment

Рассматривается симметричное ветвящееся случайное блуждание по многомерной решетке с непрерывным временем и марковским ветвящимся процессом в каждой точке решетки. Предполагается, что в начальный момент времени в точках решетки находится по одной частице и в процессе ветвления частица может произвести произвольное число потомков. Для критического ветвящегося процесса в случае невозвратного случайного блуждания по решетке доказана сходимость распределения поля частиц к предельному стационарному распределению. Показано отсутствие перемежаемости в зоне $|x-y|=O(\sqrt {t})$, где $x$, $y$ - пространственные координаты, а $t$ - время, в предположении суперэкспоненциально легких хвостов случайного блуждания и надкритичности ветвящегося процесса в точках решетки.

Symmetric branching random walks in random media: comparing theoretical and numerical results

We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous properties of a particle field, e.g., such as intermittency. However, the time intervals on which this intermittency phenomenon can be observed are very difficult to estimate in practice. In this paper, branching media containing only a finite and non-finite number of branching sources are considered. The evolution of the mean number of particles with a random point perturbation and one initial ancestor particle at a lattice point is described by an appropriate Cauchy problem for the evolutionary operator. We review some previous results about the long-time behavior of the medium-averaged moments for the particle population at every lattice point as well as the total one over the lattice and present an algorithm for the simulation of branching random walks under various assumptions about the medium, including the medium randomness. The effects arising in random non-homogeneous and homogeneous media are then compared and illustrated by simulations based on the potential with Weibull-type upper tail. A wide range of models under different assumptions on a branching medium, a configuration of branching sources, and a lattice dimension were considered during the comparison. The simulation results indicate that intermittency can be observed in random media even over finite time intervals.

Importance sampling for maxima on trees

We study the all-time supremum of the perturbed branching random walk, known to be the endogenous solution to the high-order Lindley equation: W=DmaxY,max1≤i≤N(Wi+Xi),where the {Wi} are independent copies of W, independent of the random vector (Y,N,{Xi}) taking values in R×N×R∞. Under Kesten assumptions, this solution satisfies P(W>t)∼He−αt,t→∞,where α>0 solves the Cramér–Lundberg equation E∑i=1NeαXi=1. This paper establishes the tail asymptotics of W by using the forward iterations of the map defining the fixed-point equation combined with a change of measure along a randomly chosen path. This new approach provides an explicit representation of the constant H and gives rise to unbiased and strongly efficient estimators for the rare event probabilities P(W>t).

Cover time for branching random walks on regular trees

AbstractLet 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 $r + ({2}/{\log(3/2)})\log\log r + {\mathrm{o}}(\log\log r)$ .

A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform

Given a nonincreasing null sequence \(T = (T_j)_{j \geqslant 1}\) of nonnegative random variables satisfying some classical integrability assumptions and \({\mathbb {E}}(\sum _{j}T_{j}^{\alpha })=1\) for some \(\alpha >0\), we characterize the solutions of the well-known functional equation $$\begin{aligned} f(t)\,=\,\textstyle {\mathbb {E}}\left( \prod _{j\geqslant 1}f(tT_{j})\right) ,\quad t\geqslant 0, \end{aligned}$$related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that (i) it simplifies earlier approaches in some relevant aspects; (ii) it works under weaker, close to optimal conditions in the so-called boundary case when \({\mathbb {E}}\big (\sum _{j\geqslant 1}T_{j}^{\alpha }\log T_{j}\big )=0\); (iii) it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.

Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (i.e. the set of finite words written in the alphabet {1,…,b}, with b≥2). The BRW is indexed by a critical Galton–Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, γ∈(1,2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 1∕2 to move closer to the root of Wb and probability 1∕(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)n∈N satisfying an→∞ such that the metric space (Rb(n),an−1dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

Efficient approximation of branching random walk Gibbs measures

Disordered systems such as spin glasses have been used extensively as models for high-dimensional random landscapes and studied from the perspective of optimization algorithms. In a recent paper by L. Addario-Berry and the second author, the continuous random energy model (CREM) was proposed as a simple toy model to study the efficiency of such algorithms. The following question was raised in that paper: what is the threshold $\beta_G$, at which sampling (approximately) from the Gibbs measure at inverse temperature $\beta$ becomes algorithmically hard? This paper is a first step towards answering this question. We consider the branching random walk, a time-homogeneous version of the continuous random energy model. We show that a simple greedy search on a renormalized tree yields a linear-time algorithm which approximately samples from the Gibbs measure, for every $\beta < \beta_c$, the (static) critical point. More precisely, we show that for every $\varepsilon>0$, there exists such an algorithm such that the specific relative entropy between the law sampled by the algorithm and the Gibbs measure of inverse temperature $\beta$ is less than $\varepsilon$ with high probability. In the supercritical regime $\beta > \beta_c$, we provide the following hardness result. Under a mild regularity condition, for every $\delta > 0$, there exists $z>0$ such that the running time of any given algorithm approximating the Gibbs measure stochastically dominates a geometric random variable with parameter $e^{-z\sqrt{N}}$ on an event with probability at least $1-\delta$.

Large deviations for the right-most position of a last progeny modified branching random walk

In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the $n$-th generation, which may be different from the driving increment distribution. This model was first introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Under very minimal assumptions, we derive the large deviation principle (LDP) for the right-most position of a particle in generation $n$. As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by Gantert and H\"{o}felsauer (2018).

Мартингальный метод исследования ветвящихся случайных блужданий

Мартингальный метод исследования ветвящихся случайных блужданий

Genealogy and spatial distribution of the N-particle branching random walk with polynomial tails

The N-particle branching random walk is a discrete time branching particle system with selection. We have N particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent’s location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Martingale method for studying branching random walks

Martingale method for studying branching random walks

Branching Random Walks Conditioned on Particle Numbers

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.

Barak–Erdős graphs and the infinite-bin model

A Barak-Erd\H{o}s graph is a directed acyclic version of the Erd\H{o}s-R\'enyi random graph. It is obtained by performing independent bond percolation with parameter $p$ on the complete graph with vertices $\{1,...,n\}$, in which the edge between two vertices $i<j$ is directed from $i$ to $j$. The length of the longest path in this graph grows linearly with the number of vertices, at rate $C(p)$. In this article, we use a coupling between Barak-Erd\H{o}s graphs and infinite-bin models to provide explicit estimates on $C(p)$. More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of $C$ for $p >1/2$, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of $C$ as $p \to 0$, using a coupling with branching random walks.