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This paper considers the $(n,k)$-Bernoulli--Laplace model in the case whenthere are two urns, the total number of red and white balls is the same, andthe number of selections $k$ at each step is on the same asymptotic order asthe number of balls $n$ in each urn. Our main focus is on the large-timebehavior of the corresponding Markov chain tracking the number of red balls ina given urn. Under reasonable assumptions on the asymptotic behavior of theratio $k/n$ as $n\rightarrow \infty$, cutoff in the total variation distance isestablished. A cutoff window is also provided. These results, in particular,partially resolve an open problem posed by Eskenazis and Nestoridi.

Cutoff in the Bernoulli-Laplace model with $O(n)$ swaps