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Quenched Survival of Bernoulli Percolation on Galton–Watson Trees

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Abstract

We explore the survival function for percolation on Galton–Watson trees. Letting g(T, p) represent the probability a tree T survives Bernoulli percolation with parameter p, we establish several results about the behavior of the random function \(g(\mathbf{T}, \cdot )\), where \(\mathbf{T}\) is drawn from the Galton–Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the \(k\text {th}\)-order Taylor expansion of \(g(\mathbf{T}, \cdot )\) at criticality in terms of limits of martingales defined from \(\mathbf{T}\) (this requires a moment condition depending on k); and a proof that the \(k\text {th}\) order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton–Watson tree.

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Acknowledgements

The authors would like to thank Michael Damron for drawing our attention to [6]. We would also like to thank Yuval Peres for helpful conversations.

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Authors and Affiliations

  1. University of Illinois at Chicago, Chicago, IL, USA

    Marcus Michelen

  2. University of Pennsylvania, Philadelphia, PA, USA

    Robin Pemantle

  3. University of Washington, Seattle, WA, USA

    Josh Rosenberg

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  1. Marcus Michelen
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  2. Robin Pemantle
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  3. Josh Rosenberg
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Correspondence to Marcus Michelen.

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Communicated by Pablo A Ferrari.

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Michelen, M., Pemantle, R. & Rosenberg, J. Quenched Survival of Bernoulli Percolation on Galton–Watson Trees. J Stat Phys 181, 1323–1364 (2020). https://doi.org/10.1007/s10955-020-02629-6

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  • DOI: https://doi.org/10.1007/s10955-020-02629-6

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