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There has been a lot of recent progress on branching particle systems
with selection, in particular on the $N$-particle branching random walk
($N$-BRW). In the N-BRW, $N$ particles have locations on the real line
at all times. At each time step, each of the $N$ particles has a number
of children, and each child has a random displacement from its parent’s
location. Then among the children only the $N$ rightmost are selected to
survive and reproduce in the next generation. This is a truncation
selection model.
In this talk, I will investigate the noisy version of the $N$-BRW. That
is, instead of truncation, we randomly sample $N$ particles from the
children to survive. The probability of selecting a given child depends
on its location in such a way that particles more to the right are more
likely to be selected. There are different versions of such models,
which, according to our simulations, show some similar counter-intuitive
behaviours. In this talk we will discuss explanations of these phenomena
by presenting rigorous results and conjectures on some of the noisy
selection models.
