Department of Mathematics

q-Exchangeability and the Mallows' model for random permutations

(UU)

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