The cutoff profile for random transpositions on repeated cards is asymptotically Gaussian for growing multiplicity l, with explicit forms depending on whether the number of types m is fixed or grows.
Limit profile for random transpositions.The Annals of Probability, 48(5):2323–2343, 2020
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The Cutoff Profile for Random Transpositions on Repeated Cards in the Full Range of Parameters
The cutoff profile for random transpositions on repeated cards is asymptotically Gaussian for growing multiplicity l, with explicit forms depending on whether the number of types m is fixed or grows.