Cutoff for the Bernoulli-Laplace urn model with $o(n)$ swaps
Abstract
We study the mixing time of the $(n,k)$ Bernoulli--Laplace urn model, where $k\in\{0,1,\ldots,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of the process choose uniformly at random $k$ balls from the left urn and $k$ balls from the right urn and switch them simultaneously. We show that if $k=o(n)$, this Markov chain exhibits mixing time cutoff at $\frac{n}{4k}\log n$ and window of the order $\frac{n}{k}\log\log n$. This is an extension of a classical theorem of Diaconis and Shahshahani who treated the case $k=1$.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- 10.48550/arXiv.1805.07803
- arXiv:
- arXiv:1805.07803
- Bibcode:
- 2018arXiv180507803E
- Keywords:
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- Mathematics - Probability;
- Mathematics - Combinatorics
- E-Print:
- To appear in Ann. Inst. Henri Poincar\'e Probab. Stat