scholarly journals A Note on the Transience of Critical Branching Random Walks on the Line

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Gerold Alsmeyer ◽  
Matthias Meiners
Keyword(s):  
Asymptotic Behavior ◽  
Random Walks ◽  
Lyapunov Functions ◽  
General Framework ◽  
Final Section ◽  
Branching Processes ◽  
Branching Random Walk ◽  
Branching Random Walks ◽  
International Audience ◽  
Critical Branching Random Walk

International audience Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

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.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (2) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

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 function G and a maximum principle which, we prove, is satisfied by every fixed point of G. 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.

scholarly journals Approximating Critical Parameters of Branching Random Walks

2009 ◽  
Vol 46 (02) ◽  
pp. 463-478 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Critical Parameter ◽  
Branching Random Walk ◽  
Critical Parameters ◽  
Bond Percolation ◽  
Branching Random Walks

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

scholarly journals A Note on Branching Random Walks on Finite Sets

2005 ◽  
Vol 42 (1) ◽  
pp. 287-294 ◽  
Author(s):  
Thomas Mountford ◽  
Rinaldo B. Schinazi
Keyword(s):  
Random Walk ◽  
Finite Number ◽  
Random Walks ◽  
Ecological Model ◽  
Critical Value ◽  
Strong Sense ◽  
Branching Random Walk ◽  
Finite Sets ◽  
Branching Random Walks ◽  
Number Of Species

We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

A note on the branching random walk

1982 ◽  
Vol 19 (2) ◽  
pp. 421-424 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Branching Random Walk ◽  
Branching Random Walks

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj <∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

scholarly journals An Almost-Sure Renewal Theorem for Branching Random Walks on the Line

2010 ◽  
Vol 47 (03) ◽  
pp. 811-825
Author(s):  
Matthias Meiners
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
General Characteristic ◽  
Renewal Theorem ◽  
The Real ◽  
Branching Random Walks

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.

scholarly journals Non-intersection of transient branching random walks

2020 ◽  
Vol 178 (1-2) ◽  
pp. 1-23
Author(s):  
Tom Hutchcroft
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Cayley Graph ◽  
Spectral Radius ◽  
Branching Random Walk ◽  
Expected Number ◽  
Branching Random Walks ◽  
Offspring Distribution ◽  
The Future ◽  
Supercritical Phase

Abstract Let G be a Cayley graph of a nonamenable group with spectral radius $$\rho < 1$$ ρ < 1 . It is known that branching random walk on G with offspring distribution $$\mu $$ μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $${\overline{\mu }}$$ μ ¯ satisfies $$\overline{\mu }\le \rho ^{-1}$$ μ ¯ ≤ ρ - 1 . Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase $$1<\overline{\mu } \le \rho ^{-1}$$ 1 < μ ¯ ≤ ρ - 1 , and in particular at the recurrence threshold $${\overline{\mu }} = \rho ^{-1}$$ μ ¯ = ρ - 1 , the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.

A note on the branching random walk

1982 ◽  
Vol 19 (02) ◽  
pp. 421-424
Author(s):  
Norman Kaplan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Branching Random Walk ◽  
Branching Random Walks

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj &lt;∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

scholarly journals The height of scaled attachment random recursive trees

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Luc Devroye ◽  
Omar Fawzi ◽  
Nicolas Fraiman
Keyword(s):  
Random Walks ◽  
General Class ◽  
Elementary Proof ◽  
Random Variables ◽  
Branching Random Walks ◽  
Random Node ◽  
International Audience ◽  
Recursive Trees

International audience We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.

scholarly journals An Almost-Sure Renewal Theorem for Branching Random Walks on the Line

2010 ◽  
Vol 47 (3) ◽  
pp. 811-825 ◽  
Author(s):  
Matthias Meiners
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
General Characteristic ◽  
Renewal Theorem ◽  
The Real ◽  
Branching Random Walks

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.

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