The local and global limit of the continuous-time Mallows process

The Mallows process is a process of random permutations whose marginal at time $t$ is the Mallows distribution with parameter $t$. It can be thought of as interpolating between the identity permutation and the reverse permutation on $n$ elements. We prove that
under an appropriate space and time scaling it possesses a global and a local limit. The global limit admits an explicit description in terms of the permuton limit of the Mallows distribution, analyzed by S. Starr, while the local limit is related to the construction of the
infinite Mallows distribution on $\mathbb{Z}$ due to Gnedin and Olshanskii. Joint work in progress with Radosław Adamczak.