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CATEGORIES:Probability
SUMMARY:How to initialise a second class particle? - Marto
n Balazs (Bristol)
DTSTART;TZID=Europe/London:20151117T150000
DTEND;TZID=Europe/London:20151117T160000
UID:TALK62265AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/62265
DESCRIPTION:This talk will be on interacting particle systems.
One of the best known models\nin the field is the
simple exclusion process where every site has 0 o
r 1\nparticles. It has long been established that
under certain rescaling procedure\nthis process co
nverges to solutions of a deterministic nonlinear
PDE (Burger's\nequation). Particular types of solu
tions\, called rarefaction fans\, arise from\ndecr
easing step initial data.\n\nSecond class particle
s are probabilistic objects that come from couplin
g two\ninteracting particle systems. They are very
useful and their behaviour is\nhighly nontrivial.
\n\nThe beautiful paper of P. A. Ferrari and C. Ki
pnis connects the above: they\nproved that the sec
ond class particle of simple exclusion chooses a u
niform\nrandom velocity when started in a rarefact
ion fan. The extremely elegant proof\nis based\, a
mong other ideas\, on the fact that increasing the
mean of a\nBernoulli distribution can be done by
adding or not adding 1 to the random\nvariable.\n\
nFor a long time simple exclusion was the only mod
el with an established large\nscale behaviour of t
he second class particle in its rarefaction fan. I
will\nexplain how this is done in the Ferrari-Kip
nis paper\, then show how to do this\nfor other mo
dels that allow more than one particles per site.
The main issue is\nthat most families of distribut
ions are not as nice as Bernoulli in terms of\ninc
reasing their parameter by just adding or not addi
ng 1. To overcome this we\nuse a signed\, non-prob
abilistic coupling measure that nevertheless point
s out a\ncanonical initial probability distributio
n for the second class particle. We\ncan then use
this initial distribution to greatly generalize th
e Ferrari-Kipnis\nargument. I will conclude with a
n example where the second class particle\nvelocit
y has a mixed discrete and continuous distribution
.\n\nJoint work with Attila László Nagy
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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