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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Addario-Berry, L., Ford, K.: Poisson–Dirichlet branching random walks. Ann. Appl. Prob. 23, 283–307 (2013)
Athreya, K., Ney, P.: Branching Processes. Springer, New York (1972)
Bingham, N., Doney, R.: Asymptotic properties of supercritical branching processes I: the Galton–Watson process. Adv. Appl. Prob. 6, 711–731 (1974)
Chow, Y.S., Teicher, H.: Probability Theory. Springer Texts in Statistics, 3rd edn. Springer, New York (1997). Independence, interchangeability, martingales
Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343(2), 725–745 (2016)
Fitzner, R., van der Hofstad, R.: Mean-field behavior for nearest-neighbor percolation in \(d>10\). Electron. J. Probab. 22(43), 65 (2017)
Grimmet, G.: Percolation. Grundlehren der mathematischen Wissenschaften, vol. 321, 2nd edn. Springer, New York (1999)
Hara, T., Slade, G.: Mean-field behaviour and the lace expansion. Probability and Phase Transition (Cambridge, 1993). NATO Advanced Science Institute Series C Mathematical and Physical Science, vol. 420, pp. 87–122. Kluwer Academic Publisher, Dordrecht (1994)
Harris, T.: A lower bound for the percolation probability in a certain percolation process. Proc. Camb. Philos. Soc. 56, 13–20 (1960)
Harris, T.: The Theory of Branching Processes. Springer, Berlin (1963)
Kesten, H.: The critical probability of bond percolation on the square lattice equals \({1\over 2}\). Commun. Math. Phys. 74(1), 41–59 (1980)
Kesten, H., Zhang, Y.: Strict inequalities for some critical exponents in two-dimensional percolation. J. Stat. Phys. 46(5–6), 1031–1055 (1987)
Lyons, R.: Random walks and percolation on trees. Ann. Probab. 18, 931–958 (1990)
Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56(2), 229–237 (1981)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Pablo A Ferrari.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-020-02629-6