Branching Random Walk Research Articles

Stochastic model of seed dispersal with homogeneous and non-homogeneous Poisson processes under habitat reduction conditions.

This study presents a stochastic model of seed dispersal based on a branching random walk (BRW) framework, incorporating both homogeneous and non-homogeneous Poisson point processes (PPP). Building on the model introduced by Coletti et al. (2023), we examine the effects of habitat reduction on seed dispersal dynamics. We analyze the phase transition behavior of the BRW model under varying conditions of habitat fragmentation, focusing on how these conditions influence the critical dispersal rate. Specifically, we study a BRW on the real line with a non-homogeneous PPP driven by a log-normal density, constrained between spatial barriers. Our simulations localize the critical dispersal rate with respect to barrier positions and compare this dependence between homogeneous and non-homogeneous models.

Branching random walks with regularly varying perturbations

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take the form 1/θ log X/E where θ is a positive parameter, X has an arbitrary distribution μ and E is exponential with parameter 1, independent of X. Working under finite mean assumption for μ, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when θ does not exceed certain critical parameter θ_0 . This paper complements their work by providing weak centered asymptotics for the case when θ > θ_0 and presenting the results to μ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form αn + c log n, with constants α, c depending on the ratio of θ and θ_0. We describe the limiting distribution and provide explicitly the constants appearing in the centering.

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in ℤd with a Random Environment in Time

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in ℤd with a Random Environment in Time

H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta > 0$$\\end{document}. Approaching the critical point from the transient regime, β↘βc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\searrow \\beta _{\ extrm{c}}$$\\end{document}, we show that the expected total time spent at the starting vertex diverges as ∼exp(c/β-βc)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim \\exp (c/\\sqrt{\\beta - \\beta _{\ extrm{c}}})$$\\end{document}. Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values βc<β<βcerg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{\ extrm{c}}< \\beta < \\beta _{\ extrm{c}}^{\ extrm{erg}}$$\\end{document}. In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model.

The relative multifractal analysis of a vector function in a metric space

ABSTRACT Scale invariance is a widely used concept for analysing real data from many different applications, and multifractal analysis has become the standard and powerful tool in signal processing. It characterizes the data by globally and geometrically describing the fluctuations of the local regularity. In this paper, we study the relative vectorial multifractal formalism based on vectorial multifractal Hewitt-Stromberg measures. We use a function defined on balls in a metric space and Banach valued, which is more general than measures. This leads to results on the simultaneous behaviour of possibly many local Hölder exponents. As an application, we study the relative multifractal formalism of branching random walk on the Galton-Watson tree.

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.

Derivation and travelling wave analysis of phenotype-structured haptotaxis models of cancer invasion

AbstractWe formulate haptotaxis models of cancer invasion wherein the infiltrating cancer cells can occupy a spectrum of states in phenotype space, ranging from ‘fully mesenchymal’ to ‘fully epithelial’. The more mesenchymal cells are those that display stronger haptotaxis responses and have greater capacity to modify the extracellular matrix (ECM) through enhanced secretion of matrix-degrading enzymes (MDEs). However, as a trade-off, they have lower proliferative capacity than the more epithelial cells. The framework is multiscale in that we start with an individual-based model that tracks the dynamics of single cells, which is based on a branching random walk over a lattice representing both physical and phenotype space. We formally derive the corresponding continuum model, which takes the form of a coupled system comprising a partial integro-differential equation for the local cell population density function, a partial differential equation for the MDE concentration and an infinite-dimensional ordinary differential equation for the ECM density. Despite the intricacy of the model, we show, through formal asymptotic techniques, that for certain parameter regimes it is possible to carry out a detailed travelling wave analysis and obtain invading fronts with spatial structuring of phenotypes. Precisely, the most mesenchymal cells dominate the leading edge of the invasion wave and the most epithelial (and most proliferative) dominate the rear, representing a bulk tumour population. As such, the model recapitulates similar observations into a front to back structuring of invasion waves into leader-type and follower-type cells, witnessed in an increasing number of experimental studies over recent years.

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.

Martin boundaries and asymptotic behavior of branching random walks

Martin boundaries and asymptotic behavior of branching random walks

Aggregation of states of a branching random walk over multidimensional lattice

Рассматривается случайное блуждание по многомерной решетке с непрерывным временем, которое лежит в основе ветвящегося случайного блуждания с бесконечным числом фазовых состояний. Случайное блуждание со счетным числом состояний, по которым происходит блуждание, за счет их объединения может быть сведено к системе с конечным числом состояний. Изучается асимптотическое поведение времени пребывания преобразованной системы в каждом из конечных состояний в зависимости от размерности решетки в предположении конечной дисперсии и при условии, приводящем к бесконечной дисперсии скачков исходной системы. Показано, что укрупнение состояний в рамках описанного процесса вызывает потерю марковского свойства.

Conditional central limit theorem for critical branching random walk

Conditional central limit theorem for critical branching random walk

Front propagation of branching random walk with periodic branching sources

Рассматривается модель ветвящегося случайного блуждания по целочисленной решетке $\mathbb{Z}^d$ с периодическими источниками размножения и гибели частиц. Предполагается, что режим ветвления надкритический, а для распределения скачка блуждания выполнено условие Крамера. Установлена теорема о скорости распространения фронта популяции частиц по решетке при неограниченном росте времени. Доказательства основаны на фундаментальных результатах, относящихся к пространственному распространению общего ветвящегося случайного блуждания.

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