Mixing Time of the Random-Pair Monte Shuffle

Abstract

In the random-pair Monte shuffle on $n$ cards, each step chooses two positions independently and uniformly and exchanges their cards with probability $1/2$. The chosen positions are public; the decisions to exchange cards remain hidden. We prove that the mixing time in expected conditional total variation distance is $2n\log n+O(n)$, with an exact leading constant and an $O(n)$ cutoff window. To our knowledge, this is the first public-schedule Monte shuffle whose full-permutation mixing time has been determined up to an additive $O(n)$ term. The upper bound follows by comparing two executions with the same public schedule to a classical random-transposition walk. For the lower bound, we construct a schedule-dependent test function and estimate its variance using a matrix-averaging operator. Combined with the depth bound of Czumaj and Lingas, the mixing estimate gives public switching networks with at most $2n\log n+O(n)$ switches and $O(\log n)$ depth for fixed error and failure probabilities. The author obtained this result through interaction with GPT-6 Astra.