Bootstrap percolation on Galton-Watson trees

Béla Bollobás (University of Cambridge, University of Memphis, and London Institute for Mathematical Sciences.)
Karen Gunderson (University of Bristol)
Cecilia Holmgren (Stockholm University and University of Cambridge)
Svante Janson (Uppsala University)
Michał Przykucki (University of Cambridge and London Institute for Mathematical Sciences)


Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: `infected' or `healthy'. In consecutive rounds, each healthy vertex with at least $r$ infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability $p$. In that case, given a graph $G$ and infection threshold $r$, a quantity of interest is the critical probability, $p_c(G,r)$, at which percolation becomes likely to occur. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any $b \geq r$ and for any $\epsilon > 0$ there exists a tree $T$ with branching number $\operatorname{br}(T) = b$ and critical probability $p_c(T,r) < \epsilon$. However, this is false if we limit ourselves to the well studied family of Galton--Watson trees. We show that for every $r \geq 2$ there exists a constant $c_r>0$ such that if $T$ is a Galton- Watson tree with branching number $\operatorname{br}(T) = b \geq r$ then $$p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}.$$  We also show that this bound is sharp up to a factor of $O(b)$ by giving an explicit family of Galton--Watson trees with critical probability bounded from above by $C_r e^{-\frac{b}{r-1}}$ for some constant $C_r>0$.

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Pages: 1-27

Publication Date: January 19, 2014

DOI: 10.1214/EJP.v19-2758


  • Aizenman, M.; Lebowitz, J. L. Metastability effects in bootstrap percolation. J. Phys. A 21 (1988), no. 19, 3801--3813. MR0968311
  • Balogh, József; Bollobás, Bála; Duminil-Copin, Hugo; Morris, Robert. The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc. 364 (2012), no. 5, 2667--2701. MR2888224
  • Balogh, József; Bollobás, Bála; Morris, Robert. Bootstrap percolation in high dimensions. Combin. Probab. Comput. 19 (2010), no. 5-6, 643--692. MR2726074
  • Balogh, József; Peres, Yuval; Pete, Gábor. Bootstrap percolation on infinite trees and non-amenable groups. Combin. Probab. Comput. 15 (2006), no. 5, 715--730. MR2248323
  • Balogh, József; Pittel, Boris G. Bootstrap percolation on the random regular graph. Random Structures Algorithms 30 (2007), no. 1-2, 257--286. MR2283230
  • Biskup, Marek; Schonmann, Roberto H. Metastable behavior for bootstrap percolation on regular trees. J. Stat. Phys. 136 (2009), no. 4, 667--676. MR2540158
  • J. Chalupa, P.L. Leath, and G.R. Reich, Bootstrap percolation on a Bethe latice, J. Phys. C 12 (1979), L31--L35.
  • Fontes, L. R. G.; Schonmann, R. H. Bootstrap percolation on homogeneous trees has 2 phase transitions. J. Stat. Phys. 132 (2008), no. 5, 839--861. MR2430783
  • Gautschi, Walter. Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. and Phys. 38 1959/60 77--81. MR0103289
  • K. Gunderson and M. Przykucki, Lower bounds for bootstrap percolation on Galton--Watson trees, in preparation.
  • Holroyd, Alexander E. Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 (2003), no. 2, 195--224. MR1961342
  • Janson, Svante. On percolation in random graphs with given vertex degrees. Electron. J. Probab. 14 (2009), no. 5, 87--118. MR2471661
  • Janson, Svante; Łuczak, Tomasz; Turova, Tatyana; Vallier, Thomas. Bootstrap percolation on the random graph $G_ {n,p}$. Ann. Appl. Probab. 22 (2012), no. 5, 1989--2047. MR3025687
  • Lyons, Russell. Random walks and percolation on trees. Ann. Probab. 18 (1990), no. 3, 931--958. MR1062053
  • R. Lyons and Y. Peres, Probability on trees and networks, 2012, In preparation. Current version available at

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