We prove that there exist natural generalizations of the classical bootstrap percolation model on Z 2 that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property.Van Enter (1987) (in the case d = r = 2) and Schonmann (1992) (for all d r 2) proved that r-neighbour bootstrap percolation models have trivial critical probabilities on Z d for every choice of the parameters d r 2: that is, an initial set of density p almost surely percolates Z d for every p > 0. These results effectively ended the study of bootstrap percolation on infinite lattices.Recently Bollobás, Smith and Uzzell introduced a broad class of percolation models called U-bootstrap percolation, which includes r-neighbour bootstrap percolation as a special case. They divided two-dimensional U-bootstrap percolation models into three classes -subcritical, critical and supercritical -and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical U-bootstrap percolation models have trivial critical probabilities on Z 2 . They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical U-bootstrap percolation model has a non-trivial critical probability on Z 2 . This is new except for a certain 'degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.

Consider a uniform random rooted tree on vertices labelled by [n] = {1, 2, . . . , n}, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number m ≤ n of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider m = αn and let An,α denote the event that all αn cars find spaces in the tree. Lackner and Panholzer [13] proved (via analytic combinatorics methods) that there is a phase transition in this model.1−α , whereas if α > 1/2 we have P (An,α) → 0. We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson(α) number of cars arrive at each vertex. Let X be the number of cars which visit the root of the tree. Then for α ≤ 1/2, we have E [X] ≤ 1, whereas for α > 1/2, we have E [X] = ∞. This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution.

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, pc(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 ≥ r and for any ǫ > 0 there exists a tree T with branching number br(T ) = b and critical probability pc(T, r) < ǫ. However, this is false if we limit ourselves to the well-studied family of Galton-Watson trees. We show that for every r ≥ 2 there exists a constant cr > 0 such that if T is a Galton-Watson tree with branching number br(T ) = b ≥ r then pc(T, r) > cr b e − 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 Cre − b r−1 for some constant Cr > 0. bootstrap process can be defined as follows. For any subset of verticesOften, this process is thought of as the spread of an 'infection' through the vertices of G in discrete time steps, with the vertices in one of two possible states: 'infected' or 'healthy'. For each t, A t is the set of infected vertices at time t and A is the set of vertices eventually infected when A is the set of initially infected vertices. Given a set A of initially infected vertices, percolation or complete occupation is said to occur if A = V (G).Bootstrap percolation may be thought of as a monotone version of the Glauber dynamics of the Ising model of ferromagnetism. To mimic the behaviour of ferromagnetic materials, in the classical setup, all vertices of G are assumed to belong to the set A of initially infected vertices independently with probability p. It is clear that the probability of percolation is non-decreasing in p and for a finite or infinite graph G one can define the critical probabilityfor which percolation becomes more likely to occur than not. Indeed, much work has been done in this direction for various underlying graphs and values of the infection threshold.The question of critical probability has been studied extensively in the cases of grid-like and cube-like graphs. For example, Aizenman and Lebowitz [1] showed that p c ([n] 2 , 2) decreases logarithmically with n. This was later sharpened by Holroyd [10] who showed that p c ([n] 2 , 2) = π 2 18 log n + o(1/ log n). Balogh, Bollobás, Duminil-Copin and Morris [2] generalized Holroyd's result giving a formula for p c ([n] d , r) for...