Supercritical Branching Random Walk Research Articles

Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Let {Zn}n≥0 be a d-dimensional supercritical branching random walk started from the origin. Write Zn(S) for the number of particles located in a set S⊂Rd at time n. Denote by Rn:=inf⁡{ρ≥0:Zi({|x|≥ρ})=0,∀0≤i≤n} the radius of the minimal ball (centered at the origin) containing the range of {Zi}i≥0 up to time n. In this work, we show that under some mild conditions Rn/n converges in probability to some positive constant x⁎ as n→∞. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates ofP(Rn≤xn)forx∈(0,x⁎);P(Rn≥xn)forx∈(x⁎,∞) as n→∞.

Moment Asymptotics of Population Size of Particles at Vertices for a Supercritical Branching Random Walk on a Periodic Graph

Moment Asymptotics of Population Size of Particles at Vertices for a Supercritical Branching Random Walk on a Periodic Graph

Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n≥0 on R, let Zn(A) be the number of particles located in some Borel set A⊂R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(nA)/Zn(R) converges almost surely to ν(A) as n→∞, where ν(⋅) is the standard Gaussian measure on R. Large deviation probabilities of Zn(nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate ofP(Zn(nA)/Zn(R)≥ν(A)+Δn) as n→∞, where {Δn}n≥0 is some positive sequence tending to zero.

Moment asymptotics of particle numbers at vertices for a supercritical branching random walk on a periodic graph

Рассматривается надкритическое симметричное ветвящееся случайное блуждание по многомерному графу с непрерывным временем и источниками генерации частиц, расположенными периодически. Получена логарифмическая асимптотика для моментов численностей частиц в каждой вершине графа при $t\to\infty$.

Lower Deviation Probabilities for Level Sets of the Branching Random Walk

Given a supercritical branching random walk $$\{Z_n\}_{n\ge 0}$$ on $${\mathbb {R}}$$ , let $$Z_n(A)$$ be the number of particles located in a set $$A\subset {\mathbb {R}}$$ at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for $$\theta \in [0,1)$$ , $$n^{-1}\log Z_n([\theta x^* n,\infty ))$$ converges almost surely to $$\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)$$ as $$n\rightarrow \infty $$ , where $$x^*$$ is the speed of the maximal position of $$\{Z_n\}_{n\ge 0}$$ and $$I(\cdot )$$ is the large deviation rate function of the underlying random walk. In this work, we investigate its lower deviation probabilities, in other words, the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( Z_n([\theta x^* n,\infty ))<e^{an}\right) \end{aligned}$$ as $$n\rightarrow \infty $$ , where $$a\in [0,\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*))$$ . Our results complete those in Chen and He (Ann Institut Henri Poincare Probab Stat 56:2507–2539, 2020), Gantert and Höfelsauer (Electron Commun Probab 23(34):1–12, 2018) and Öz (Latin Am J Probab Math Stat 17:711–731, 2020).

On the Variance of the Particle Number of a Supercritical Branching Random Walk on Periodic Graphs

We study the asymptotic behavior of the variance of the local particle number for a supercritical branching random walk with branching sources that are located periodically on Zd.

Fluctuations of Biggins’ martingales at complex parameters

Le comportement en temps long d’une marche aléatoire branchante surcritique peut être décrit et analysé en utilisant les martingales de Biggins, à paramètres réels ou complexes. L’étude de ces martingales prises en des paramètres complexes est un sujet d’étude assez récent. En supposant que certaines conditions pour leur convergence vers une limite non-dégénérée sont vérifiées, nous étudions les fluctuations de ces martingales autour de leurs limites. Nous observons trois régimes différents. D’abord, nous montrons que dans une région dans laquelle les paramètres sont de petite norme, les fluctuations sont gaussiennes, et les lois limites sont des mélanges de variables aléatoires gaussiennes réelles ou complexes. Nous obtenons également le comportement au bord de cette région. Dans un second temps, nous trouvons une région dans l’espace des paramètres dans laquelle les fluctuations des martingales sont déterminées par les valeurs extrêmes dans la marche aléatoire branchante. Finalement, il existe une région critique (typiquement sur le bord de l’ensemble des paramètres pour lesquels les martingales convergent vers une limite non-dégénérée) où les fluctuations sont de type stable, et les lois limites sont les lois de valeurs en un temps aléatoire de processus de Lévy satisfaisant des propriétés d’invariance similaires à la stabilité.

On the derivative martingale in a branching random walk

We work under the Aïdékon–Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that EZ1{Z≤x}=logx+o(logx) as x→∞. Also, we provide necessary and sufficient conditions under which EZ1{Z≤x}=logx+const+o(1) as x→∞. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of, at most, linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.

Fluctuations of the Propagation Front of a Catalytic Branching Walk

We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z^d (d is positive integer). The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M_n of the particles at time n. They proved that M_n/n converges to \mu almost surely (on a set of local non-degeneracy of CBRW), as n tends to infinity, where \mu>0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations of M_n with respect to \mu n, as n grows to infinity. Bulinskaya (2018) extended the strong limit theorem by Carmona and Hu having estimated the rate of the population propagation for the front of a multidimensional CBRW. Now our aim is to analyze fluctuations of the propagation front in CBRW on Z^d. We not only solve the problem in a multidimensional setting but also, treating the case of an arbitrary finite number of catalysts for d = 1, generalize the result by Carmona and Hu with the help of other probabilistic-analytic methods. Keywords and phrases: catalytic branching random walk, supercritical regime, spread of population, propagation front, fluctuations of front.

Isotropic Multidimensional Catalytic Branching Random Walk with Regularly Varying Tails

The study completes a series of the author’s works devoted to the spread of particles population in supercritical catalytic branching random walk (CBRW) on a multidimensional lattice. The CBRW model describes the evolution of a system of particles combining their random movement with branching (reproduction and death) which only occurs at fixed points of the lattice. The set of such catalytic points is assumed to be finite and arbitrary. In the supercritical regime the size of population, initiated by a parent particle, increases exponentially with positive probability. The rate of the spread depends essentially on the distribution tails of the random walk jump. If the jump distribution has “light tails”, the “population front”, formed by the particles most distant from the origin, moves linearly in time and the limiting shape of the front is a convex surface. When the random walk jump has independent coordinates with a semiexponential distribution, the population spreads with a power rate in time and the limiting shape of the front is a star-shape nonconvex surface. So far, for regularly varying tails (“heavy” tails), we have considered the problem of scaled front propagation assuming independence of components of the random walk jump. Now, without this hypothesis, we examine an “isotropic” case, when the rate of decay of the jumps distribution in different directions is given by the same regularly varying function. We specify the probability that, for time going to infinity, the limiting random set formed by appropriately scaled positions of population particles belongs to a set $B$ containing the origin with its neighborhood, in $\mathbb{R}^d$. In contrast to the previous results, the random cloud of particles with normalized positions in the time limit will not concentrate on coordinate axes with probability one.

Random walk on barely supercritical branching random walk

Let $${\mathcal {T}}$$ be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean $$\mu >1$$, conditioned to survive. Let $$\varphi _{\mathcal {T}}$$ be a random embedding of $${\mathcal {T}}$$ into $${\mathbb {Z}}^d$$ according to a simple random walk step distribution. Let $${\mathcal {T}}_p$$ be percolation on $${\mathcal {T}}$$ with parameter p, and let $$p_c = \mu ^{-1}$$ be the critical percolation parameter. We consider a random walk $$(X_n)_{n \ge 1}$$ on $${\mathcal {T}}_p$$ and investigate the behavior of the embedded process $$\varphi _{{\mathcal {T}}_p}(X_n)$$ as $$n\rightarrow \infty $$ and simultaneously, $${\mathcal {T}}_p$$ becomes critical, that is, $$p=p_n\searrow p_c$$. We show that when we scale time by $$n/(p_n-p_c)^3$$ and space by $$\sqrt{(p_n-p_c)/n}$$, the process $$(\varphi _{{\mathcal {T}}_p}(X_n))_{n \ge 1}$$ converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.

Operator Equations of Branching Random Walks

Consideration is given to the continuous-time supercritical branching random walk over a multidimensional lattice with a finite number of particle generation sources of the same intensity both with and without constraint on the variance of jumps of random walk underlying the process. Asymptotic behavior of the Green function and eigenvalue of the evolution operator of the mean number of particles under source intensity close to the critical one was established.

On Lp-convergence of the Biggins martingale with complex parameter

We prove necessary and sufficient conditions for the Lp-convergence, p>1, of the Biggins martingale with complex parameter in the supercritical branching random walk. The results and their proofs are much more involved (especially in the case p∈(1,2)) than those for the Biggins martingale with real parameter. Our conditions are ultimate in the case p≥2 only.

A Limit Theorem for Supercritical Random Branching Walks with Branching Sources of Varying Intensity

We consider a supercritical symmetric continuous-time branching random walk on a multidimensional lattice with a finite number of particle generation sources of varying positive intensities without...

Stable-like fluctuations of Biggins’ martingales

Let (Wn(θ))n∈N0 be Biggins’ martingale associated with a supercritical branching random walk, and let W(θ) be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of W1(θ) belongs to the domain of normal attraction of an α-stable distribution for some α∈(1,2), then, as n→∞, there is weak convergence of the tail process (W(θ)−Wn−k(θ))k∈N0, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with α-stable marginals.

On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

Given a branching random walk on $${\mathbb {R}}$$ started from the origin, where the tail of the branching law decays at least exponentially fast and the offspring number is at least one, let $$Z_n(\cdot )$$ be the counting measure which counts the number of individuals at the n-th generation located in a given set. Under some mild conditions, it is known (Biggins in Stoch. Process. Appl. 34:255–274, 1990) that for any interval $$A\subset {\mathbb {R}}$$ , $$\frac{Z_n(\sqrt{n}A)}{Z_n({\mathbb {R}})}$$ converges a.s. to $$\nu (A)$$ , where $$\nu $$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( \frac{Z_n(\sqrt{n}A)}{Z_n({\mathbb {R}})}-\nu (A)>\Delta \right) , \end{aligned}$$ for $$\Delta \in (0, 1-\nu (A))$$ . We consider both the Schroder case, where the offspring number could be one, and the Bottcher case, where the offspring number is at least two.

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form Vn−φlogn+oP(logn), where Vn is a function of the environment that behaves as a random walk and φ>0 is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.

The near-critical Gibbs measure of the branching random walk

Considerons une marche aleatoire branchante surcritique reelle dans le cas frontiere et la mesure de Gibbs associee $\nu_{n,\beta}$ sur la $n$-ieme generation, qui est aussi la mesure de polymere sur un arbre desordonne avec temperature inverse $\beta$. La convergence de la fonction de partition $W_{n,\beta}$, apres renormalisation, vers une limite non-triviale a ete demontree par Aidekon et Shi (Ann. Probab. 42 (3) (2014) 959–993) dans le cas critique $\beta=1$ et par Madaule (J. Theoret. Probab. 30 (1) (2017) 27–63) pour $\beta>1$. On s’interesse ici au cas presque-critique, ou $\beta_{n}\to1$, et on montre la convergence de $W_{n,\beta_{n}}$, apres renormalisation, vers la limite de la martingale derivee a un facteur multiplicatif pres. D’autre part, les trajectoires de particules tirees selon la mesure de Gibbs $\nu_{n,\beta}$ ont ete etudiees par Madaule (Stochastic Process. Appl. 126 (2) (2016) 470–502) dans le cas critique, avec convergence vers le meandre brownien, et par Chen, Madaule et Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) dans le regime de desordre fort, avec convergence vers l’excursion brownienne. On montre ici la convergence des trajectoires de particules tirees selon la mesure de Gibbs presque-critique et cela fait apparaitre une famille continue de processus allant du meandre jusqu’a l’excursion ou jusqu’au mouvement brownien.

Spread of a catalytic branching random walk on a multidimensional lattice

For a supercritical catalytic branching random walk on Zd, d∈N, with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. It is shown that in the result of the proper normalization of the particles positions in the limit there are a.s. no particles outside the closed convex surface in Rd which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation.

Convergence of complex martingales in the branching random walk: the boundary

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda $ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda $ of $\Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda $ and require weaker moment assumptions than the ones used by Biggins.