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 profiles for reversible Markov chains.Probability Theory and Related Fields, 182(1):157–188, 2022
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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.