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We consider a generalization of the Bernoulli-Laplace model in which thereare two urns and $n$ total balls, of which $r$ are red and $n - r$ white, andwhere the left urn holds $m$ balls. At each time increment, $k$ balls arechosen uniformly at random from each urn and then swapped. This system can beused to model phenomena such as gas particle interchange between containers orcard shuffling. Under a reasonable set of assumptions, we bound the mixing timeof the resulting Markov chain asymptotically in $n$ with cutoff at $\log{n}$and constant window. Among other techniques, we employ the spectral analysis ofarXiv:0906.4242 on the Markov transition kernel and the chain coupling tools ofarXiv:2203.08647 and arXiv:1606.01437.

Cutoff in the Bernoulli-Laplace Model With Unequal Colors and Urn Sizes