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

Continuous-time Mallows processes are processes of random permutations of the set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution with parameter $t$. Recently Corsini showed that there exists a unique Markov Mallows process whose left inversions are independent counting processes. We prove that this process admits a global and a local limit as $n \to \infty$. The global limit, obtained after suitably rescaling space and time, is an explicit stochastic process on $[0,1]$ whose description is based on the permuton limit of the Mallows distribution, analyzed by Starr. The local limit is a process of permutations of $\mathbb{Z}$ which is closely related to the construction of the Mallows distribution on permutations of $\mathbb{Z}$ due to Gnedin and Olshanski. Our results demonstrate an analogy between the asymptotic behavior of Mallows processes and the recently studied limiting properties of random sorting networks.