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CATEGORIES:Probability
SUMMARY:Fortuin-Kastelyn type representations for Threshol
d Gaussian and Stable Vectors: aka Divide and Col
or processes (joint work with Malin Palö Forsström
) - Jeff Steif (Chalmers)
DTSTART;TZID=Europe/London:20191015T140000
DTEND;TZID=Europe/London:20191015T150000
UID:TALK131455AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/131455
DESCRIPTION:We consider the following simple model: one starts
with a finite (or countable) set V\,\n a random p
artition of V and a parameter p in [0\,1]. The "Ge
neralized Divide and Color Model"\n is the {0\,1}-
valued process indexed by V obtained by independen
tly\, for each partition element\n in the random p
artition chosen\, with probability p assigning all
the elements 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. M
any models fall into this context:\n (1) the 0 ext
ernal field Ising model (where the random partitio
n is given by FK percolation)\,\n (2) the stationa
ry distributions for the voter model (where the ra
ndom partition is given by\n coalescing random wal
ks)\, (3) random walk in random scenery and (4)\n
the original "Divide and Color Model" introduced a
nd studied by Olle Häggström. In earlier work\, to
gether with Johan Tykesson\, we studied what one c
ould say about such processes. In joint work with
Malin Palö Forsström\, we study the question of wh
ich threshold Gaussian\n and stable vectors have s
uch a representation: (A threshold Gaussian (stabl
e) vector is a vector\n obtained by taking a Gauss
ian (stable) vector and a threshold h and looking
where\n the vector exceeds the threshold h). The a
nswer turns out to be quite varied depending\n on
properties of the vector and the threshold\; it tu
rns out that h=0 behaves quite\n differently than
h different from 0. Among other results\, in the l
arge h regime\, we obtain a\n phase transition in
the stability exponent alpha for stable vectors an
d the critical value is\n alpha=1/2. I will also b
riefly describe some related results by Forsström
concerning such questions for\n the Ising Model wi
th a nonzero external field.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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