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

TL;DR

This paper extends the Mallows model to a two-sided infinite setting for permutations of integers, analyzing its properties and differences from the one-sided case.

Contribution

It introduces a new two-sided infinite permutation distribution extending the Mallows model and studies its key features and symmetries.

Findings

Analysis of symmetries of the distribution

Characterization of the support and marginal distributions

Comparison with the one-sided infinite case

Abstract

We introduce a probability distribution Q on the group of permutations of the set Z of integers. 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 the set N of natural numbers, was studied previously in our paper [Gnedin and Olshanski, Ann. Prob. 38 (2010), 2103-2135; arXiv:0907.3275]. We analyze various features of Q such as its symmetries, the support, and the marginal distributions.