Randomisations, coheir sequences and NSOP1
[Joint with A Chernikov and N Ramsey]
Recall that if T is a theory, then its Keisler randomisation, T^R, is the theory of spaces of random variables which take values in a model of T .
It was show some time ago that if T has IP (e.g., simple unstable), then T^R has TP2, and in particular not simple.
In Eilat I announced the following result [with Chernikov and Ramsey] :
A. If T is NSOP1, then its randomisation T^R is NSOP1
B. If T is simple (so T^R is NSOP1), then we have the following characterisation of Kim independence over models in T^R : a \ind_M b if and only if a and b are "probabilistically independent" over M as well as independent with probability one in the sense of T .
Today, some improvements to the argument allow us to prove B also when T is merely NSOP1.
I shall try to explain the notions involved and the main idea of the proof of the last statement.