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SUMMARY:How to initialise a second class particle? - Marton Balazs (Bristo
 l)
DTSTART:20151117T150000Z
DTEND:20151117T160000Z
UID:TALK62265@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:This talk will be on interacting particle systems. One of the 
 best known models\nin the field is the simple exclusion process where ever
 y site has 0 or 1\nparticles. It has long been established that under cert
 ain rescaling procedure\nthis process converges to solutions of a determin
 istic nonlinear PDE (Burger's\nequation). Particular types of solutions\, 
 called rarefaction fans\, arise from\ndecreasing step initial data.\n\nSec
 ond class particles are probabilistic objects that come from coupling two\
 ninteracting particle systems. They are very useful and their behaviour is
 \nhighly nontrivial.\n\nThe beautiful paper of P. A. Ferrari and C. Kipnis
  connects the above: they\nproved that the second class particle of simple
  exclusion chooses a uniform\nrandom velocity when started in a rarefactio
 n fan. The extremely elegant proof\nis based\, among 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 model with an established large\nscale behaviour of
  the second class particle in its rarefaction fan. I will\nexplain how thi
 s is done in the Ferrari-Kipnis paper\, then show how to do this\nfor othe
 r models that allow more than one particles per site. The main issue is\nt
 hat most families of distributions are not as nice as Bernoulli in terms o
 f\nincreasing their parameter by just adding or not adding 1. To overcome 
 this we\nuse a signed\, non-probabilistic coupling measure that neverthele
 ss points out a\ncanonical initial probability distribution for the second
  class particle. We\ncan then use this initial distribution to greatly gen
 eralize the Ferrari-Kipnis\nargument. I will conclude with an example wher
 e the second class particle\nvelocity has a mixed discrete and continuous 
 distribution.\n\nJoint work with Attila László Nagy
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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