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36
BOOTSTRAP PERCOLATION IN HIGH DIMENSIONS
"... Abstract. In rneighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vert ..."
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Abstract. In rneighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n] d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability pc([n] d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d ≫ log n. The bootstrap process has been extensively studied on [n] d when d is a fixed constant and 2 � r � d, and in these cases pc([n] d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2] d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n] d, d) asymptotically if d � (log log n) 2+ε, and gave much sharper bounds for the hypercube. Here we prove the following result: let λ be the smallest positive root of the equation k=0 so λ ≈ 1.166. Then 16λ d2 () log d
Bootstrap percolation on the random graph Gn,p
, 2010
"... Bootstrap percolation on the random graph Gn,p is a process of spread of “activation ” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least r ≥ 2 active neighbours become active as well. We study t ..."
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Bootstrap percolation on the random graph Gn,p is a process of spread of “activation ” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least r ≥ 2 active neighbours become active as well. We study the size A ∗ of the final active set. The parameters of the model are, besides r (fixed) and n (tending to ∞), the size a = a(n) of the initially active set and the probability p = p(n) of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model the final size of activation with a high probability is either n − o(n) or it is o(n). We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for A ∗ ; we also prove a central limit theorem for A ∗ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.
A sharper threshold for bootstrap percolation in two dimensions
"... Twodimensional bootstrap percolation is a cellular automaton in which sites become ‘infected ’ by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially inf ..."
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Twodimensional bootstrap percolation is a cellular automaton in which sites become ‘infected ’ by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp firstorder approximation: pc ∼ π2 /(18 log n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is −(log n) −3/2+o(1) , and moreover determining it up to a poly(log log n)factor.
LINEAR ALGEBRA AND BOOTSTRAP PERCOLATION
, 1107
"... Abstract. In Hbootstrap percolation, a set A ⊂ V(H) of initially ‘infected ’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the Hbootstrap process, in which H encodes copies of H in a graph G. We find the min ..."
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Abstract. In Hbootstrap percolation, a set A ⊂ V(H) of initially ‘infected ’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the Hbootstrap process, in which H encodes copies of H in a graph G. We find the minimum size of a set A that leads to complete infection when G and H are powers of complete graphs and H encodes induced copies of H in G. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent to (edge) Hbootstrap percolation on a complete graph. 1.
Probability and Geometry on Groups  Lecture notes for a graduate course
, 2015
"... These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley grap ..."
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These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley graphs of infinite groups; 2) the algebraic properties of these groups; and 3) the behaviour of probabilistic processes (most importantly, random walks, harmonic functions, and percolation) on these Cayley graphs. I try to be as little abstract as possible, emphasizing examples rather than presenting theorems in their most general forms. I also try to provide guidance to recent research literature. In particular, there are presently over 150 exercises and many open problems that might be accessible to PhD students. It is also hoped that researchers working either in probability or in geometric group theory will find these notes useful to enter the other field.
PERCOLATION SINCE SAINTFLOUR
"... There has been a great deal of interest and activity in percolation theory since the two SaintFlour courses, [76, 111], of 1984 and 1996 reprinted in this new edition. We present here a summary of progress since the first publications of our lecture notes. ..."
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There has been a great deal of interest and activity in percolation theory since the two SaintFlour courses, [76, 111], of 1984 and 1996 reprinted in this new edition. We present here a summary of progress since the first publications of our lecture notes.