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CATEGORIES:Combinatorics Seminar
SUMMARY:Bootstrap Percolation in the Hypercube - Natasha M
orrison (Oxford)
DTSTART;TZID=Europe/London:20151022T143000
DTEND;TZID=Europe/London:20151022T153000
UID:TALK60767AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60767
DESCRIPTION:The \\emph{$r$-neighbour bootstrap process} on a g
raph $G$ starts with an initial set of ``infected'
' vertices and\, at each step of the process\, a h
ealthy vertex becomes infected if it has at least
$r$ infected neighbours (once a vertex becomes inf
ected\, it remains infected forever). If every ver
tex of $G$ becomes infected during the process\, t
hen we say that the initial set \\emph{percolates}
. \n\nIn this talk I will discuss the proof of a c
onjecture of Balogh and Bollob\\'{a}s: for fixed $
r$ and $d\\to\\infty$\, the minimum cardinality of
a percolating set in the $d$-dimensional hypercub
e is $\\frac{1+o(1)}{r}\\binom{d}{r-1}$. One of th
e key ideas behind the proof exploits a connection
between bootstrap percolation and weak saturation
. This is joint work with Jonathan Noel.\n
LOCATION:MR12
CONTACT:Andrew Thomason
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