A weakly supercritical mode in a branching random walk

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.     

Large deviations for a symmetric branching random walk on a multidimensional lattice

An important role in the theory of branching random walks is played by the problem of the spectrum of a bounded symmetric operator, the generator of a random walk on a multidimensional integer lattice, with a one-point potential. We consider operators with potentials of a more general form that take nonzero values on a finite set of points of the integer lattice. The resolvent analysis of such operators has allowed us to study branching random walks with large deviations. We prove limit theorems on the asymptotic behavior of the Green function of transition probabilities. Special attention is paid to the case when the spectrum of the evolution operator of the mean numbers of particles contains a single eigenvalue. The results obtained extend the earlier studies in this field in such directions as the concept of a reaction front and the structure of a population inside a front and near its boundary.     

Spectral properties of evolutionary operators in branching random walk models

We introduce a model of continuous-time branching random walk on a finite-dimensional integer lattice with finitely many branching sources of three types and study the spectral properties of the operator describing the evolution of the mean numbers of particles both at an arbitrary source and on the entire lattice. For the leading positive eigenvalue of the operator, we obtain existence conditions determining exponential growth in the number of particles in this model.     

Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

Hitting times with taboo for a random walk

For a symmetric homogeneous and irreducible random walk on the d-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in [0,??]) determined by a starting point x, a hitting state y, and a taboo state z. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on ? ^d except for a simple random walk on ?, the order of the distribution tail decrease is specified by dimension d only. In contrast, for a simple random walk on ?, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points x, y, and z. These problems originated in recent study of a branching random walk on ? ^d with a single source of branching.     

The monotonicity of the probability of return into the source in models of branching random walks

The paper discusses two models of a branching random walk on a many-dimensional lattice with birth and death of particles at a single node being the source of branching. The random walk in the first model is assumed to be symmetric. In the second model an additional parameter is introduced which enables “artificial” intensification of the prevalence of branching or walk at the source and, as the result, violating the symmetry of the random walk. The monotonicity of the return probability into the source is proved for the second model, which is a key property in the analysis of branching random walks.     

Limiting laws of supercritical branching random walks

In this Note, we make explicit the limit law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in Barral et al. (2012) [5] for a continuous analogue of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectures formulated by physicists (Derrida and Spohn, 1988 [9]) about the Poisson–Dirichlet nature of the jumps in the limit, and precising the conjecture by giving the spatial distribution of these jumps.     

Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law

We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥?1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d =?1 and d =?2.

     

Operator Equations of Branching Random Walks

Methodology and Computing in Applied Probability - Consideration is given to the continuous-time supercritical branching random walk over a multidimensional lattice with a finite number of particle...     

The survival of branching annihilating random walk

Summary Branching annihilating random walk is an interacting particle system on . As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown here that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non-attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by Gray and Griffeath is used.     

On the Mean Number of Particles of a Branching Random Walk on ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with Periodic Sources of Branching

We consider a continuous-time branching random walk on ? ^d , where the particles are born and die on a periodic set of points (sources of branching). The spectral properties of the evolution operator for the mean number of particles at an arbitrary point of ? ^d are studied. This operator is proved to have a positive spectrum, which leads to an exponential asymptotic behavior of the mean number of particles as t → ∞.     

Random Walks on Directed Covers of Graphs

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.     

Critical survival barrier for branching random walk

We consider a branching random walk with an absorbing barrier, where the associated one-dimensional random walk is in the domain of attraction of an α-stable law. We shall prove that there is a barrier and a critical value such that the process dies under the critical barrier, and survives above it. This generalizes previous result in the case that the associated random walk has finite variance.     

Recurrence and transience of multitype branching random walks

We study a discrete time Markov process with particles being able to perform discrete time random walks and create new particles, known as branching random walk (BRW). We suppose that there are particles of different types, and the transition probabilities, as well as offspring distribution, depend on the type and the position of the particle. Criteria of (strong) recurrence and transience are presented, and some applications (spatially homogeneous case, Lamperti BRW, many-dimensional BRW) are studied.     

Limit distributions for the number of particles in branching random walks

We study branching random walks with continuous time. Particles performing a random walk on ?², are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.     

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

On finiteness of discrete spectrum of three-particle Schrödinger operator on a lattice

On three-dimensional lattice we consider a system of three quantum particles (two of them are identical (fermions) and the third one is of another nature) that interact with the help of paired short-range gravitational potentials. We prove the finiteness of a number of bound states of respective Schrödinger operator in a case, when potentials satisfy some conditions and zero is a regular point for two-particle sub-Hamiltonian. We find a set of values for particles masses values such that Schrödinger operator may have only finite number of eigenvalues lying to the left of essential spectrum.     

Multidimensional Watson lemma and its applications

We prove the multidimensional analog of the well-knownWatson lemma and then apply it to prove a local limit theorem for the transition probabilities of symmetric random walks on the multidimensional lattice with infinite variance of jumps.     

Eigenvalues of Quantum Walks of Grover and Fourier Types

A necessary and sufficient conditions for a certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of \(\pm \, 1\) as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has 1 as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general conditions for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics, such as Grover walks on the triangular lattice, products and deformations of Grover walks, are also discussed.

     

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.