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

Continuous-time Mallows processes are processes of random permutations of the set \1, , 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. 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 Z which is closely related to the construction of the Mallows distribution on permutations of 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.