2024/04/12 by Radosław Adamczak, Adamczak, Radosław, Michał Kotowski +1
Mathematics · #05A05 #60C05 #60J27 #60J75 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2404.08554
openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.