The two-sided infinite extension of the Mallows model for random permutations

We introduce a probability distribution Q on the infinite group SZ of permutations of the set of integers Z. The distribution Q is a natural extension of the Mallows distribution on the finite symmetric group. A one-sided infinite counterpart of Q, supported by the group of permutations of N, was studied previously in our paper [A. Gnedin, G. Olshanski, q-Exchangeability via quasi-invariance, Ann. Probab. 38 (2010) 2103–2135, arXiv:0907.3275]. We analyze various features of Q such as its symmetries, the support, and the marginal distributions.