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SUMMARY:Scale-free percolation - van der Hofstad\, R (Technische Universit
 t Eindhoven)
DTSTART:20150320T100000Z
DTEND:20150320T110000Z
UID:TALK58507@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-authors: Mia Deijfen (Stockholm University)\, Gerard Hooghi
 emstra (Delft University of Technology)\n\nWe propose and study a random g
 raph model on the hypercubic lattice that interpolates between models of s
 cale-free random graphs and long-range percolation.\n\nIn our model\, each
  vertex $x$ has a weight $W_x$\, where the weights of different vertices a
 re i.i.d. random variables. Given the weights\, the edge between $x$ and $
 y$ is\, independently of all other edges\, occupied with probability $1-{m
 athrm{e}}^{-lambda W_xW_y/|x-y|^{lpha}}$\, where\n\n(a) $lambda$ is the p
 ercolation parameter\,\n(b) $|x-y|$ is the Euclidean distance between $x$ 
 and $y$\, and\n(c) $lpha$ is a long-range parameter.\n\nThe most interest
 ing behavior can be observed when the random weights have a power-law dist
 ribution\, i.e.\, when $mathbb{P}(W_x>w)$ is regularly varying with expone
 nt $1-	au$ 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 
 cluster as $gamma$ varies. Our results interpolate between those proved in
  inhomogeneous random graphs\, where a wealth of further results is known\
 , and those in long-range percolation. We also discuss many open problems\
 , inspired both by recent work on long-range percolation (i.e.\, $W_x=1$ f
 or every $x$)\, and on inhomogeneous random graphs (i.e.\, the model on th
 e complete graph of size $n$ and where $|x-y|=n$ for every $x\neq y$).\n
LOCATION:Seminar Room 1\, Newton Institute
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