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SUMMARY:Fortuin-Kastelyn type representations for Threshold Gaussian and  
 Stable Vectors: aka Divide and Color processes (joint work with Malin Pal
 ö Forsström) - Jeff Steif (Chalmers)
DTSTART:20191015T130000Z
DTEND:20191015T140000Z
UID:TALK131455@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:We consider the following simple model: one starts with a fini
 te (or countable) set V\,\n a random partition of V and a parameter p in [
 0\,1]. The "Generalized Divide and Color Model"\n is the {0\,1}-valued pro
 cess indexed by V obtained by independently\, for each partition element\n
  in the random partition chosen\, with probability p assigning all the ele
 ments of the partition\n element the value 1\, and with probability 1−p\
 , assigning all the elements of the partition\n element the value 0. Many 
 models fall into this context:\n (1) the 0 external field Ising model (whe
 re the random partition is given by FK percolation)\,\n (2) the stationary
  distributions for the voter model (where the random partition is given by
 \n coalescing random walks)\, (3) random walk in random scenery and (4)\n 
 the original "Divide and Color Model" introduced and studied by Olle Hägg
 ström. In earlier work\, together with Johan Tykesson\, we studied what o
 ne could say about such processes. In joint work with Malin Palö Forsstr
 öm\, we study the question of which threshold Gaussian\n and stable vecto
 rs have such a representation: (A threshold Gaussian (stable) vector is a 
 vector\n obtained by taking a Gaussian (stable) vector and a threshold h a
 nd looking where\n the vector exceeds the threshold h). The answer turns o
 ut to be quite varied depending\n on properties of the vector and the thre
 shold\; it turns out that h=0 behaves quite\n differently than h different
  from 0. Among other results\, in the large h regime\, we obtain a\n phase
  transition in the stability exponent alpha for stable vectors and the cri
 tical value is\n alpha=1/2. I will also briefly describe some related resu
 lts by Forsström concerning such questions for\n the Ising Model with a n
 onzero external field.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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