Branching Random Walk Research Articles

Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.

Scaling Limit of the Path Leading to the Leftmost Particle in a Branching Random Walk

We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the $n$th generation behaves asymptotically like $\frac{3}{2}\,\log n$, provided the nonextinction of the system. The main goal of this paper is to prove that the path from the root to the leftmost particle, after a suitable normalization, converges weakly to a Brownian excursion in $D([0,1],{\bf R})$.

From Derrida's random energy model to branching random walks: from 1 to 3

We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter $\alpha \in \left[0,1\right]$: choosing $\alpha=0$ yields the random energy model by Derrida (REM), whereas $\alpha=1$ corresponds to the branching random walk (BRW). When the parameter $\alpha$ increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as $\alpha<1$ strictly, the limiting extremal process is always Poissonian.

The one-arm exponent for mean-field long-range percolation

Consider a long-range percolation model on $\mathbb{Z}^d$ where the probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open is proportional to $\|x-y\|_2^{-d \alpha}$ for some $\alpha \gt 0$ and where $d \gt 3 \min\{2,\alpha\}$. We prove that in this case the one-arm exponent equals $\min\{4,\alpha\}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $\alpha$ when $\alpha =4$.

Complete Classification of Catalytic Branching Processes

We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may only occur at the points of catalysis. The results obtained generalize and strengthen those known in cases of CBP with a single catalyst and of branching random walk on d-dimensional integer lattice with a finite number of sources of particles generation. We propose to classify CBP with N catalysts as supercritical, critical or subcritical according to the value of the Perron root of a specified NxN matrix. Such classification agrees with the moment analysis performed here for local and total particles numbers. By introducing the criticality set C we also consider the influence of catalysts parameters on the process behavior. The proof is based on construction of auxiliary multi-type Bellman-Harris processes with the help of hitting times under taboo and on application of multidimensional renewal theorems. Keywords and phrases: catalytic branching process, classification, hitting times under taboo, moment analysis, multi-type Bellman-Harris process.

A probabilistic model for martensitic avalanches

We present a probabilistic model for the description of martensitic avalanches. Our approach to the analysis of the model is based on an associated general branching random walk process. Comparisons are reported for numerical and analytical solutions and experimental observations.

Phenomenological picture of fluctuations in branching random walks.

We propose a picture of the fluctuations in branching random walks, which leads to predictions for the distribution of a random variable that characterizes the position of the bulk of the particles. We also interpret the 1/sqrt[t] correction to the average position of the rightmost particle of a branching random walk for large times t≫1, computed by Ebert and Van Saarloos, as fluctuations on top of the mean-field approximation of this process with a Brunet-Derrida cutoff at the tip that simulates discreteness. Our analytical formulas successfully compare to numerical simulations of a particular model of a branching random walk.

Maximums on trees

We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R=D(⋁i=1NCiRi)∨Q, where (Q,N,C1,C2,…) is a random vector with N∈N∪{∞}, P(|Q|>0)>0 and nonnegative weights {Ci}, and {Ri}i∈N is a sequence of i.i.d. copies of R independent of (Q,N,C1,C2,…); =D denotes equality in distribution. Furthermore, when Q>0 this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of R is power-law, i.e., P(|R|>x)∼Hx−α, for some α>0 and H>0. This has direct implications for the tail behavior of other well known branching recursions.

THE RANGE OF TREE-INDEXED RANDOM WALK

We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.

Coupling Limit Order Books and Branching Random Walks

We consider a model for a one-sided limit order book proposed by Lakner, Reed and Stoikov (2013). We show that it can be coupled with a branching random walk and use this coupling to answer a nontrivial question about the long-term behavior of the price. The coupling relies on a classical idea of enriching the state space by artificially creating a filiation, in this context between orders of the book, which we believe has the potential of being useful for a broader class of models.

Coupling Limit Order Books and Branching Random Walks

We consider a model for a one-sided limit order book proposed by Lakner, Reed and Stoikov (2013). We show that it can be coupled with a branching random walk and use this coupling to answer a nontrivial question about the long-term behavior of the price. The coupling relies on a classical idea of enriching the state space by artificially creating a filiation, in this context between orders of the book, which we believe has the potential of being useful for a broader class of models.

Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x) +\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is H\"{o}lder continuous of index $\gamma>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

CPMC-Lab: A Matlab package for Constrained Path Monte Carlo calculations

We describe CPMC-Lab, a Matlab program for the constrained-path and phaseless auxiliary-field Monte Carlo methods. These methods have allowed applications ranging from the study of strongly correlated models, such as the Hubbard model, to ab initio calculations in molecules and solids. The present package implements the full ground-state constrained-path Monte Carlo (CPMC) method in Matlab with a graphical interface, using the Hubbard model as an example. The package can perform calculations in finite supercells in any dimensions, under periodic or twist boundary conditions. Importance sampling and all other algorithmic details of a total energy calculation are included and illustrated. This open-source tool allows users to experiment with various model and run parameters and visualize the results. It provides a direct and interactive environment to learn the method and study the code with minimal overhead for setup. Furthermore, the package can be easily generalized for auxiliary-field quantum Monte Carlo (AFQMC) calculations in many other models for correlated electron systems, and can serve as a template for developing a production code for AFQMC total energy calculations in real materials. Several illustrative studies are carried out in one- and two-dimensional lattices on total energy, kinetic energy, potential energy, and charge- and spin-gaps. Program summaryProgram title: CPMC-LabCatalogue identifier: AEUD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2850No. of bytes in distributed program, including test data, etc.: 24838Distribution format: tar.gzProgramming language:Matlab.Computer: The non-interactive scripts can be executed on any computer capable of running Matlab with all Matlab versions. The GUI requires Matlab R2010b (version 7.11) and above. Operating system: Windows, Mac OS X, Linux.RAM: Variable.Classification: 7.3.External routines:MatlabNature of problem:Obtaining ground state energy of a repulsive Hubbard model in a supercell in any number of dimensions.Solution method:In the Constrained Path Monte Carlo (CPMC) method, the ground state of a many-fermion system is projected from an initial trial wave function by a branching random walk in an overcomplete basis of Slater determinants. Constraining the determinants according to a trial wave function |ΨT〉 removes the exponential decay of the signal-to-noise ratio characteristic of the sign problem. The method is exact if |ΨT〉 is exact.Unusual features:Direct and interactive environment with a Graphical User Interface for beginners to learn and study the Constrained Path Monte Carlo method with minimal overhead for setup.Running time:The sample provided takes a few seconds to run, the batch sample a few minutes.

Universal properties of branching random walks in confined geometries

Characterizing the occupation statistics of random walks through confined geometries amounts to assessing the distribution of the travelled length ℓ and the number of collisions n performed by the stochastic process in a given region, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this letter, we derive two key results: first, we show that such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, and have thus a universal character; second, we obtain a stronger version of these formulas relating the travelled length density and the collision density at any point of the phase space. Our results are key to such technological issues as the analysis of radiation flow for nuclear reactor design and medical diagnosis and apply more broadly to physical and biological systems with diffusion, reproduction and death.

Branching random walks with random environments in time

We consider a branching random walk on ℝ with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let \(\tilde Z_n (t)\) be its Laplace transform. We show the convergence of the free energy n−1log \(\tilde Z_n (t)\), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale \(\tilde Z_n (t)/\mathbb{E}[\tilde Z_n (t)|\xi ]\).

Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in $${\mathbb{R}^d}$$ R d

Consider an $\R^d$-valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets $E(K)$ of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points $K$. This goes beyond multifractal analysis, which only considers those level sets when $K$ ranges in the set of singletons $\{\alpha\}$, $\alpha\in\R^d$. We also give a $0$-$\infty$ law for the Hausdorff and packing measures of the level sets $E(\{\alpha\})$, and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels $\alpha$ which do not belong to the range of the gradient of the free energy. This covers in particular a situation until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or $L^q$-spectrum), when a first order phase transition occurs.

Electrical Resistance of the Low Dimensional Critical Branching Random Walk

We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d < 6 is O(n 1-α ) for some universal constant α > 0. This answers a question of Barlow et al. (Commun Math Phys 278:385–431, 2008).

Strong Local Survival of Branching Random Walks is Not Monotone

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Strong Local Survival of Branching Random Walks is Not Monotone

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

On the maximal displacement of critical branching random walk

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance $\eta^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $\sigma^{2}$ and finite third moment then the distribution of the rightmost position $M$ reached by a particle of the branching random walk satisfies $P\{M \geq x\}\sim 6\eta^{2}/ (\sigma^{2}x^{2})$ as $x \rightarrow \infty$. We also prove a conditional limit theorem for the distribution of the rightmost particle location at time $n$ given that the process survives for $n$ generations.