Approximating Critical Parameters of Branching Random Walks

Bertacchi, Daniela; Zucca, Fabio

doi:10.1017/s0021900200005581

Cited by 16 publications

(56 citation statements)

References 25 publications

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“…Our main result answers both questions regarding λ s : for quasi-transitive BRWs on Z d the strong critical parameter coincides with the one on C ∞ (this result was actually already in [11], Theorem 7.1, but here we provide a different proof which can be extended to answer the second question). Moreover the sequence of the strong critical parameters of k-type contact processes restricted to C ∞ converge to the one of the BRW on Z d .…”

supporting

confidence: 57%

“…We observe that the equality lim k→∞ λ k s (Z d , µ) = λ s (Z d , µ) has already been proven in [11], Theorem 5.1. The result for the weak critical parameter can be obtained when λ w (Z d , µ) = λ s (Z d , µ), which is, for instance, true when µ is quasi-transitive and symmetric; see Section 2.…”

supporting

confidence: 53%

“…We restate here both the lemma and the approximation theorem. It is worth noting that the irreducibility assumptions which were present in [11,32,34] are here dropped.…”

mentioning

confidence: 99%

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Branching random walks and multi-type contact-processes on the percolation cluster of $\mathbb{Z}^{d}$

Bertacchi¹,

Zucca²

2015

Ann. Appl. Probab.

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In this paper we prove that, under the assumption of quasitransitivity, if a branching random walk on Z d survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster C∞ of a supercritical Bernoulli percolation. When no more than k individuals per site are allowed, we obtain the k-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already k individuals are present. We prove that local survival of the branching random walk on Z d also implies that for k sufficiently large the associated k-type contact process survives on C∞. This implies that the strong critical parameters of the branching random walk on Z d and on C∞ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated k-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.

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confidence: 57%

supporting

confidence: 53%

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Branching random walks and multi-type contact-processes on the percolation cluster of $\mathbb{Z}^{d}$

Bertacchi¹,

Zucca²

2015

Ann. Appl. Probab.

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“…When such an upper bound is fixed, say at most m particles per site, we get the m-type contact process. The branching random walk can be obtained as the limiting process as m goes to infinity ( [4,8,23]). of modified BRWs.…”

Section: Introductionmentioning

confidence: 99%

A generating function approach to branching random walks

Bertacchi¹,

Zucca²

2017

Braz. J. Probab. Stat.

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It is well known that the behaviour of a branching process is completely described by\ud the generating function of the offspring law and its fixed points. Branching random walks are a natural\ud generalization of branching processes: a branching process can be seen as a one-dimensional\ud branching random walk. We define a multidimensional generating function associated to a given\ud branching random walk. The present paper investigates the similarities and the differences of the\ud generating functions, their fixed points and the implications on the underlying stochastic process,\ud between the one-dimensional (branching process) and the multidimensional case (branching random\ud walk). In particular, we show that the generating function of a branching random walk can\ud have uncountably many fixed points and a fixed point may not be an extinction probability, even\ud in the irreducible case (extinction probabilities are always fixed points). Moreover, the generating\ud function might not be a convex function. We also study how the behaviour of a branching random\ud walk is affected by local modifications of the process. As a corollary, we describe a general\ud procedure by which we can modify a continuous-time branching random walk which has a weak\ud phase and turn it into a continuous-time branching random walk which has strong local survival\ud for large or small values of the parameter and non-strong local survival for intermediate values of\ud the parameter

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“…Recently, some authors in the literature of branching random walks have defined local survival, meaning that, for every given type i and arbitrarily large epoch T , there is at least one individual of type i alive at some time t > T , with global survival, meaning that at least one individual is alive at any time, and strong local survival, when the two have the same probability. We refer the reader to [1], [2], and [17].…”

Section: Introductionmentioning

confidence: 99%

Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

2013

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We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

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Approximating Critical Parameters of Branching Random Walks

Cited by 16 publications

References 25 publications

Branching random walks and multi-type contact-processes on the percolation cluster of $\mathbb{Z}^{d}$

Branching random walks and multi-type contact-processes on the percolation cluster of $\mathbb{Z}^{d}$

A generating function approach to branching random walks

Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

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