Lower bounds for bootstrap percolation on Galton-Watson trees

Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.


Introduction
Bootstrap percolation, a type of cellular automaton, was introduced by Chalupa, Leath and Reich [1] and has been used to model a number of physical processes. Given a graph G and threshold r ≥ 2, the r-neighbour bootstrap process on G is defined as follows: Given A ⊆ V (G), set A 0 = A and for each t ≥ 1, define where N (v) is the neighbourhood of v in G. The closure of a set A is A = t≥0 A t . Often the bootstrap process is thought of as the spread, in discrete time steps, of an 'infection' on a graph. Vertices are in one of two states: 'infected' or 'healthy' and a vertex with at least r infected neighbours becomes itself infected, if it was not already, at the next time step. For each t, the set A t is the set of infected vertices at time t. A set A ⊆ V (G) of initially infected vertices is said to Usually, the behaviour of bootstrap processes is studied in the case where the initially infected vertices, i.e., the set A, are chosen independently at random with a fixed probability p. For an infinite graph G the critical probability is defined by This is different from the usual definition of critical probability for finite graphs, which is generally defined as the infimum of the values of p for which percolation is more likely to occur than not.
In this paper, we consider bootstrap percolation on Galton-Watson trees and answer a conjecture in [3] on lower bounds for their critical probabilities. For any offspring distribution ξ on N ∪ {0}, let T ξ denote a random Galton-Watson tree with offspring distribution ξ. For any fixed offspring distribution ξ, the critical probability p c (T ξ , r) is almost surely a constant (see Lemma 3.2 in [3]) and we shall give lower bounds on the critical probability in terms of various moments of ξ.
Bootstrap processes on infinite regular trees were first considered by Chalupa, Leath and Reich [1]. Later, Balogh, Peres and Pete [2] studied bootstrap percolation on arbitrary infinite trees and one particular example of a random tree given by a Galton-Watson branching process. In [3], Galton-Watson branching processes were further considered, and it was shown that for every r ≥ 2, there is a constant c r > 0 so that r − 1 and in addition, for every α ∈ (0, 1], there is a positive constant c r,α so that, Additionally, in [3] it was conjectured that for any r ≥ 2, inequality (1) holds for any α ∈ (0, r − 1]. As our main result, we show that this conjecture is true. For the proofs to come, some notation from [3] is used. If an offspring distribution ξ is such that P(ξ < r) > 0, then one can easily show that p c (T ξ , r) = 1. With this in mind, for r-neighbour bootstrap percolation, we only consider offspring distributions with ξ ≥ r almost surely.
Definition 1. For every r ≥ 2 and k ≥ r, define and for any offspring distribution ξ with ξ ≥ r almost surely, define Some facts, which can be proved by induction, about these functions are used in the proofs to come. For any r ≥ 2, we have g r r (x) = r−1 i=0 (1 − x) i and for any k > r, Hence, for all distributions ξ we have G r . Developing a formulation given by Balogh, Peres and Pete [2], it was shown in [3] (see Theorem 3.6 in [3]) that if ξ ≥ r, then

Results
In this section, we shall prove a family of lower bounds on the critical probability p c (T ξ , r) based on the (1 + α)-moments of the offspring distributions ξ for all α ∈ (0, r − 1], using a modification of the proofs of Lemmas 3.7 and 3.8 in [3] together with some properties of the gamma function and the beta function. Recall that the gamma function is given, for z with ℜ(z) > 0, by Γ(z) = ∞ 0 t z−1 e −t dt and for all n ∈ N, satisfies Γ(n) = (n − 1)!. The beta function is given, for ℜ(x), ℜ(y) > 0, by Γ(x+y) . We shall use the following bounds on the ratio of two values of the gamma function obtained by Gautschi [4]. For n ∈ N and 0 ≤ s ≤ 1 we have Let us now state our main result.
Combining equation (8) with equation (7) yields For every natural number n ∈ [1, r − 2], note that lim α→n − c r,α > 0 and, by the monotone convergence theorem, there is a constant c r,n > 0 so that This completes the proof of the lemma.
In the above proof, as α → (r − 1) − , c 1 (r, α) → ∞ and hence lim α→(r−1) − c r,α = 0, so the proof of Lemma 3 does not directly extend to the case α = r − 1. We deal with this problem in the next lemma. Using a different approach we prove an essentially best possible lower bound on p c (T ξ , r) based on the r-th moment of the distribution ξ. The sharpness of our bound is demonstrated by the b-branching tree T b , a Galton-Watson tree with a constant offspring distribution, for which, as a function of b, we have p c (T b , r) = (1 + o(1))(1 − 1/r) (r−1)! b r 1/(r−1) (see Lemma 3.7 in [3]).
Proof. As in the proof of Lemma 3.7 of [3] note that for every k ≥ r and t ∈ [0, 1], Using the lower bound in inequality (9) for the function G r ξ (x) yields Since the maximum value of G r ξ (x) is at least as big as G r ξ (1 − t 0 ), by equation (3), .
This completes the proof of the lemma.
Theorem 2 now follows immediately from Lemmas 3 and 4. It is not possible to extend a result of the form of Theorem 2 to α > r − 1, as demonstrated, again, by the regular b-branching tree. For every α, the (1 + α)-th moment of this distribution is b 1+α and the critical probability for the constant distribution is . As we already noted, Lemma 4 is asymptotically sharp, giving the best possible constant in Theorem 2 for any r ≥ 2 and α = r − 1. We now show that for α ∈ (0, r − 1), Theorem 2 is also best possible, up to constants. In [3], it was shown that for every r ≥ 2, there is a constant C r such that if b ≥ (r − 1) log(4er), then there is an offspring distribution η r,b with E[η r,b ] = b and p c (T η r,b , r) ≤ C r e − b r−1 . It was shown that there are k 1 = k 1 (r, b) ≤ e(r − 2)e b r−1 − 1 and A, λ ∈ (0, 1) so that the distribution η r,b is given by P(η r,b = k) =      r−1 k(k−1) r < k ≤ k 1 , k = 2r + 1 1 r + λA k = r r−1 (2r+1)2r + (1 − λ)A k = 2r + 1.