Matches in SemOpenAlex for { <https://semopenalex.org/work/W4300522014> ?p ?o ?g. }
Showing items 1 to 66 of
66
with 100 items per page.
- W4300522014 abstract "In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A subset 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 -> infinity. The main question is to determine the critical probability p_c([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 gg log n. The bootstrap process has been extensively studied on [n]^d when d is a fixed constant and 2 leq r leq d, and in these cases p_c([n]^d,r) has recently been determined up to a factor of 1 + o(1) as n -> infinity. At the other end of the scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant factor, and Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if d > (log log n)^{2+eps}, and gave much sharper bounds for the hypercube. Here we prove the following result: let lambda be the smallest positive root of the equation sum_{k=0}^infty (-1)^k lambda^k / (2^{k^2-k} k!) = 0, so lambda approx 1.166. Then (16lambda / d^2) (1 + (log d / sqrt{d})) 2^{-2sqrt{d}} < p_c([2]^d,2) < (16lambda / d^2) (1 + (5(log d)^2 / sqrt{d})) 2^{-2sqrt{d}} if d is sufficiently large, and moreover we determine a sharp threshold for the critical probability p_c([n]^d,2) for every function n = n(d) with d gg log n." @default.
- W4300522014 created "2022-10-03" @default.
- W4300522014 creator A5015732625 @default.
- W4300522014 creator A5059910732 @default.
- W4300522014 creator A5079041086 @default.
- W4300522014 date "2009-07-17" @default.
- W4300522014 modified "2023-09-25" @default.
- W4300522014 title "Bootstrap percolation in high dimensions" @default.
- W4300522014 doi "https://doi.org/10.48550/arxiv.0907.3097" @default.
- W4300522014 hasPublicationYear "2009" @default.
- W4300522014 type Work @default.
- W4300522014 citedByCount "0" @default.
- W4300522014 crossrefType "posted-content" @default.
- W4300522014 hasAuthorship W4300522014A5015732625 @default.
- W4300522014 hasAuthorship W4300522014A5059910732 @default.
- W4300522014 hasAuthorship W4300522014A5079041086 @default.
- W4300522014 hasBestOaLocation W43005220141 @default.
- W4300522014 hasConcept C114614502 @default.
- W4300522014 hasConcept C118615104 @default.
- W4300522014 hasConcept C120665830 @default.
- W4300522014 hasConcept C121332964 @default.
- W4300522014 hasConcept C134306372 @default.
- W4300522014 hasConcept C169760540 @default.
- W4300522014 hasConcept C199360897 @default.
- W4300522014 hasConcept C2777027219 @default.
- W4300522014 hasConcept C2778113609 @default.
- W4300522014 hasConcept C2780457167 @default.
- W4300522014 hasConcept C33923547 @default.
- W4300522014 hasConcept C34388435 @default.
- W4300522014 hasConcept C41008148 @default.
- W4300522014 hasConcept C50820777 @default.
- W4300522014 hasConcept C7321624 @default.
- W4300522014 hasConcept C86803240 @default.
- W4300522014 hasConceptScore W4300522014C114614502 @default.
- W4300522014 hasConceptScore W4300522014C118615104 @default.
- W4300522014 hasConceptScore W4300522014C120665830 @default.
- W4300522014 hasConceptScore W4300522014C121332964 @default.
- W4300522014 hasConceptScore W4300522014C134306372 @default.
- W4300522014 hasConceptScore W4300522014C169760540 @default.
- W4300522014 hasConceptScore W4300522014C199360897 @default.
- W4300522014 hasConceptScore W4300522014C2777027219 @default.
- W4300522014 hasConceptScore W4300522014C2778113609 @default.
- W4300522014 hasConceptScore W4300522014C2780457167 @default.
- W4300522014 hasConceptScore W4300522014C33923547 @default.
- W4300522014 hasConceptScore W4300522014C34388435 @default.
- W4300522014 hasConceptScore W4300522014C41008148 @default.
- W4300522014 hasConceptScore W4300522014C50820777 @default.
- W4300522014 hasConceptScore W4300522014C7321624 @default.
- W4300522014 hasConceptScore W4300522014C86803240 @default.
- W4300522014 hasLocation W43005220141 @default.
- W4300522014 hasLocation W43005220142 @default.
- W4300522014 hasOpenAccess W4300522014 @default.
- W4300522014 hasPrimaryLocation W43005220141 @default.
- W4300522014 hasRelatedWork W2037300822 @default.
- W4300522014 hasRelatedWork W2042064591 @default.
- W4300522014 hasRelatedWork W2154745345 @default.
- W4300522014 hasRelatedWork W2551359169 @default.
- W4300522014 hasRelatedWork W2739495308 @default.
- W4300522014 hasRelatedWork W2963875036 @default.
- W4300522014 hasRelatedWork W2972075590 @default.
- W4300522014 hasRelatedWork W3100688885 @default.
- W4300522014 hasRelatedWork W4310889339 @default.
- W4300522014 hasRelatedWork W1584735299 @default.
- W4300522014 isParatext "false" @default.
- W4300522014 isRetracted "false" @default.
- W4300522014 workType "article" @default.