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How to initialise a second class particle?

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If you have a question about this talk, please contact Perla Sousi.

This talk will be on interacting particle systems. One of the best known models in the field is the simple exclusion process where every site has 0 or 1 particles. It has long been established that under certain rescaling procedure this process converges to solutions of a deterministic nonlinear PDE (Burger’s equation). Particular types of solutions, called rarefaction fans, arise from decreasing step initial data.

Second class particles are probabilistic objects that come from coupling two interacting particle systems. They are very useful and their behaviour is highly nontrivial.

The beautiful paper of P. A. Ferrari and C. Kipnis connects the above: they proved that the second class particle of simple exclusion chooses a uniform random velocity when started in a rarefaction fan. The extremely elegant proof is based, among other ideas, on the fact that increasing the mean of a Bernoulli distribution can be done by adding or not adding 1 to the random variable.

For a long time simple exclusion was the only model with an established large scale behaviour of the second class particle in its rarefaction fan. I will explain how this is done in the Ferrari-Kipnis paper, then show how to do this for other models that allow more than one particles per site. The main issue is that most families of distributions are not as nice as Bernoulli in terms of increasing their parameter by just adding or not adding 1. To overcome this we use a signed, non-probabilistic coupling measure that nevertheless points out a canonical initial probability distribution for the second class particle. We can then use this initial distribution to greatly generalize the Ferrari-Kipnis argument. I will conclude with an example where the second class particle velocity has a mixed discrete and continuous distribution.

Joint work with Attila László Nagy

This talk is part of the Probability series.

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