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SUMMARY:Phase transition in loop percolation - Dr Artem Sapozhnikov\, Max-
 Planck Institut fur Mathematik\, Leipzig
DTSTART:20140527T153000Z
DTEND:20140527T163000Z
UID:TALK51893@talks.cam.ac.uk
CONTACT:37296
DESCRIPTION:We study properties of clusters formed by a Poisson ensemble o
 f Markovian loops on $Z^d$ ($d\\geq 3$). The intensity measure of the ense
 mble is proportional to a parameter $\\alpha$ which measures the amount of
  loops entering the picture. First\, we observe a non-trivial percolation 
 phase transition with respect to $\\alpha$. Then\, we focus on the tails o
 f the\ndistributions of the diameter and the size of clusters in the subcr
 itical regime\, which decay at most polynomially through the whole domain.
  We show\nthat for $d\\geq 5$\, if the cluster of the origin is large\, th
 en it will typically contain very close to the origin a loop comparable in
  diameter and\nin size with the whole cluster. On the other hand\, for the
  dimensions $3$ and $4$\, if the cluster of the origin is large\, then typ
 ically\, there are no\nlarge loops near the origin. This phenomenon affect
 s values of various critical exponents: they do not depend on $\\alpha$ if
  the dimension is at\nleast 5\, but do depend on $\\alpha$ if the dimensio
 n is $3$. We conclude with some open questions. Joint work with Yinshan Ch
 ang (MPI Leipzig).
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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