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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Scale-free percolation - van der Hofstad\, R (Tech
nische Universitt Eindhoven)
DTSTART;TZID=Europe/London:20150320T100000
DTEND;TZID=Europe/London:20150320T110000
UID:TALK58507AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58507
DESCRIPTION:Co-authors: Mia Deijfen (Stockholm University)\, G
erard Hooghiemstra (Delft University of Technology
)\n\nWe propose and study a random graph model on
the hypercubic lattice that interpolates between m
odels of scale-free random graphs and long-range p
ercolation.\n\nIn our model\, each vertex $x$ has
a weight $W_x$\, where the weights of different ve
rtices are i.i.d. random variables. Given the weig
hts\, the edge between $x$ and $y$ is\, independen
tly of all other edges\, occupied with probability
$1-{mathrm{e}}^{-lambda W_xW_y/|x-y|^{lpha}}$\,
where\n\n(a) $lambda$ is the percolation parameter
\,\n(b) $|x-y|$ is the Euclidean distance between
$x$ and $y$\, and\n(c) $lpha$ is a long-range par
ameter.\n\nThe most interesting behavior can be ob
served when the random weights have a power-law di
stribution\, i.e.\, when $mathbb{P}(W_x>w)$ is reg
ularly varying with exponent $1- au$ for some $ au
>1$. In this case\, we see that the degrees are in
finite a.s. when $gamma =lpha( au-1)/d leq 1$ or
$lphaleq d$\, while the degrees have a power-law
distribution with exponent $gamma$ when $gamma>1$.
\n\nOur main results describe phase transitions in
the positivity of the percolation critical value
and in the graph distances in the percolation clus
ter as $gamma$ varies. Our results interpolate bet
ween those proved in inhomogeneous random graphs\,
where a wealth of further results is known\, and
those in long-range percolation. We also discuss m
any open problems\, inspired both by recent work o
n long-range percolation (i.e.\, $W_x=1$ for every
$x$)\, and on inhomogeneous random graphs (i.e.\,
the model on the complete graph of size $n$ and w
here $|x-y|=n$ for every $x\neq y$).\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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