Universiteit Utrecht

Department of Mathematics

q-Exchangeability and the Mallows' model for random permutations

Sasha Gnedin

24 September 2009

The q-exchangeability is introduced as quasi-invariance under permutations, with a special cocycle. For 0< q<1 , the property means that the transposition of adjacent letters uw multiplies the probability by q or 1/q, depending on whether u < w or u > w, respectively.

A q-analogue of de Finetti's theorem is proved for the real-valued sequences. In contrast to the classical case with q=1, the order on the reals plays for the q-analogues a significant role.

An explicit construction of ergodic q-exchangeable measures involves a random shuffling of the set N={1,2,..} by iteration of the geometric choice, which is an analogue of Mallows' measure on finite permutations.

We establish the classical limit, and connect the q-exchangeability to a class of increasing lattice walks and to multitype asymmetric exclusion processes.

Joint work with Grigori Olshanski.

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Alexandra Babenko