Refinements of stochastic domination

Broman, Erik I.; Häggström, Olle; Steif, Jeffrey E.

doi:10.1007/s00440-006-0496-1

Cited by 4 publications

(7 citation statements)

References 15 publications

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“…It is claimed in [7] that this assumption cannot be weakened much; our results show that if one adds the very natural condition (3), then in fact one can significantly weaken the assumptions.…”

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process W...mentioning

confidence: 68%

“…The first one appears (without proof) in [3]. The proof we give is due to Olle Häggström (unpublished).…”

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process W...mentioning

confidence: 94%

“…This result also leads to continuity of the percolation function above p c for random trees. Corollary 1.7 Consider an inhomogeneous Galton-Watson process satisfying (1), (2) and (3). Then the percolation function θ(p) is continuous above p c (T ).…”

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process mentioning

confidence: 98%

“…However, it is the case that there is a subsequence along which θ n is bounded away from 0. Proposition 1.4 Consider an inhomogeneous Galton-Watson process with offspring distributions {L n } ∞ n=1 satisfying (1), (2) and (3). Then there exists a sequence {n k } ∞ k=1 of increasing integers and a constant C > 0 such that for all k ≥ 1, θ n k ≥ C.…”

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process mentioning

confidence: 99%

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Survival of inhomogeneous Galton-Watson processes

Broman

Meester

2008

Advances in Applied Probability

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We study survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an a.s. constant. We also shed some light on the way the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parametrized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the n-th level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalisations of results by Lyons (1992).

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Section: Proposition 12 For Any Inhomogeneous Galton-watson Process W...mentioning

confidence: 68%

“…The first one appears (without proof) in [3]. The proof we give is due to Olle Häggström (unpublished).…”

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process W...mentioning

confidence: 94%

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process mentioning

confidence: 98%

Section: Proposition 12 For Any Inhomogeneous Galton-watson Process mentioning

confidence: 99%

See 2 more Smart Citations

Survival of inhomogeneous Galton-Watson processes

Broman

Meester

2008

Advances in Applied Probability

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“…The first one appears (without proof) in [2]. The proof we give is due to Olle Häggström (unpublished).…”

Section: Proofs Of Theorem 12 and Proposition 13mentioning

confidence: 99%

Survival of inhomogeneous Galton-Watson processes

Broman

Meester

2008

Adv. Appl. Probab.

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We study the survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an almost sure constant. We also shed some light on the way in which the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parameterized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the nth level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalizations of results in Lyons (1992).

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Dynamical stability of percolation for some interacting particle systems and ɛ-movability

Broman¹,

Steif²

2006

Ann. Probab.

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In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward ε-movability which will be a key tool for our analysis.[15] was whether there could exist atypical times at which the percolation structure looks different than at a fixed time.We record here some of the results from [15]; (i) for any graph

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Refinements of stochastic domination

Cited by 4 publications

References 15 publications

Survival of inhomogeneous Galton-Watson processes

Survival of inhomogeneous Galton-Watson processes

Survival of inhomogeneous Galton-Watson processes

Dynamical stability of percolation for some interacting particle systems and ɛ-movability

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