Branching Random Walk Research Articles

Limit theorems for a branching random walk in a random or varying environment

We consider a branching random walk on the real line with a stationary and ergodic environment (ξn) indexed by time, in which a particle of generation n gives birth to a random number of particles of the next generation, which move on the real line; the joint distribution of the number of children and their displacements on the real line depends on the environment ξn at time n. Let Zn be the counting measure at time n, which counts the number of particles of generation n situated in a Borel set of the real line. For the case where the corresponding branching process is supercritical, we establish limit theorems such as large and moderate deviation principles, central and local limit theorems on the counting measures Zn, convergence of the free energy, law of large numbers on the leftmost and rightmost positions at time n, and the convergence to infinite divisible laws. The varying environment case is also considered.

1-stable fluctuation of the derivative martingale of branching random walk

In this paper, we study the functional convergence in law of the fluctuations of the derivative martingale of branching random walk on the real line. Our main result strengthens the results of Buraczewski et al. (2021) and is the branching random walk counterpart of the main result of Maillard and Pain (2019) for branching Brownian motion.

Branching Random Walks in a Random Killing Environment with a Single Reproduction Source

We consider a continuous-time branching random walk on Z in a random non-homogeneous environment. The process starts with a single particle at initial time t=0. This particle can walk on the lattice points or disappear with a random intensity until it reaches the certain point, which we call the reproduction source. At the source, the particle can split into two offspring or jump out of the source. The offspring of the initial particle evolves according to the same law, independently of each other and the entire prehistory. The aim of this paper is to study the conditions for the presence of exponential growth of the average number of particles at every lattice point. For this purpose, we investigate the spectrum of the random evolution operator of the average particle numbers. We derive the condition under which there is exponential growth with probability one. We also study the process under the violation of this condition and present the lower and upper estimates for the probability of exponential growth.

Conditional central limit theorem for critical branching random walk

Conditional central limit theorem for critical branching random walk

On the barrier problem of branching random walk in a time-inhomogeneous random environment

On the barrier problem of branching random walk in a time-inhomogeneous random environment

On periodic branching random walks on $\mathbf{Z}^d$ with infinite variance of jumps

В работе рассматривается периодическое ветвящееся случайное блуждание с периодически расположенными источниками ветвления. Предполагается, что интенсивности переходов случайного блуждания удовлетворяют нескольким условиям симметрии, а на переходные интенсивности случайного блуждания накладывается условие, приводящее к бесконечной дисперсии скачков. В этом случае получен старший член асимптотики средней численности частиц в произвольной точке решетки при больших временах.

Convergence in law for the capacity of the range of a critical branching random walk

Convergence in law for the capacity of the range of a critical branching random walk

Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree

We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.

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→∞.

Interfacing Branching Random Walks with Metropolis Sampling: Constraint Release in Auxiliary-Field Quantum Monte Carlo.

We present an approach to interface branching random walks with Markov chain Monte Carlo sampling and to switch seamlessly between the two. The approach is discussed in the context of auxiliary-field quantum Monte Carlo (AFQMC) but can be applied to other Monte Carlo calculations or simulations. In AFQMC, the formulation of branching random walks along imaginary-time is needed to realize a constraint to control the sign or phase problem. The constraint is derived from an exact gauge condition and is in practice implemented approximately with a trial wave function or trial density matrix, which can break exactness in the algorithm. We use the generalized Metropolis algorithm to sample a selected portion of the imaginary-time path after it has been produced by the branching random walk. This interfacing allows a constraint release to follow seamlessly from constrained-path sampling, which can reduce the systematic error from the latter. It also provides a way to improve the computation of correlation functions and observables that do not commute with the Hamiltonian. We illustrate the method in atoms and molecules, where improvements in accuracy can be clearly quantified and near-exact results are obtained. We also discuss the computation of the variance of the Hamiltonian and propose a convenient way to evaluate it stochastically without changing the scaling of AFQMC.

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

A branching random walk in the presence of a hard wall

AbstractWe consider a branching random walk on a d-ary tree of height n ( $n \in \mathbb{N}$ ), in the presence of a hard wall which restricts each value to be positive, where d is a natural number satisfying $d\geqslant2$ . We consider the behaviour of Gaussian processes with long-range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of vertices. We observe a relation with the expected maximum of the processes. We find the probability of the event that the branching random walk is positive at every vertex in the nth generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$ .

Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let ∂T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of ∂T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on ∂T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=t∈∂T:SnX(t)−αSnX˜(t)∼snasn→+∞. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension.

Large deviations of extremes in branching random walk with regularly varying displacements

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the n-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the n-th generation.

The maximum of a branching random walk with stretched exponential tails

Nous étudions une marche aléatoire branchante uni-dimensionelle quand les déplacements n’ont pas des moments exponentiels. Plus précisement, la queue d’un déplacement X se comporte comme suit : P[X>t]∼aexp{−λtr}, pour des constantes a,λ>0 et r∈(0,1). Nous donnons une description détaillée du comportement asymptotique du maximum, en montrant des lois limites presque sûres, des theorèmes de convergence en loi et une dichotomie basée sur une condition de croissance. Ces lois limites diverses font apparaître des différences interéssantes entre les deux régimes r∈(0,2/3) et r∈(2/3,1), et le cas critique r=2/3 est encore différent.

Free energy subadditivity for symmetric random Hamiltonians

We consider a random Hamiltonian H:Σ→R defined on a compact space Σ that admits a transitive action by a compact group G. When the law of H is G-invariant, we show its expected free energy relative to the unique G-invariant probability measure on Σ, which obeys a subadditivity property in the law of H itself. The bound is often tight for weak disorder and relates free energies at different temperatures when H is a Gaussian process. Many examples are discussed, including branching random walks, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum Sherrington–Kirkpatrick and Sachdev–Ye–Kitaev models.

The Spread Speed of Multiple Catalytic Branching Random Walks

The Spread Speed of Multiple Catalytic Branching Random Walks

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

Branching random walk with infinite progeny mean: A tale of two tails

We study the extremes of branching random walks under the assumption that the underlying Galton–Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the nth generation when the tail of the displacement behaves like exp(−K(x)), where either K is a regularly varying function of index r>0, or K has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when r>1.

On the role of reduced habitat in the phase transition of a stochastic model for seed dispersal

Habitat loss is one of the biggest threats facing plant species nowadays. We formulate a simple mathematical model of seed dispersal on reduced habitats to discuss survival of the species in relation to the habitat size and seeds production rate. Seeds get dispersed around the mother plant via several agents in a random way. In our model, seeds landing sites are distributed according to a homogeneous Poisson point process with a constant rate on . We will assume that each seed will successfully germinate and grow into a new plant with the same characteristics as the mother plant. The time is discrete, scaled according to generations of plants, or can represent years, since annual plants go through an entire growing cycle during 1 year. Then we will assume there are two symmetric barriers with respect to the origin and consider that the growth cannot evolve past the barriers. Imposing barriers correspond to the physical limitation of the habitat. We appeal to tools of probability theory to formalize and study such a model, which can be seen as a discrete‐time one‐dimensional branching random walk with barriers. By means of coupling techniques and the comparison with suitably constructed multitype branching processes, we localize the critical parameter of the process around which there is survival with positive probability or extinction almost surely. In addition, we consider a discrete‐space version of the model for which exact results are also obtained.