Limit Profile for the Bernoulli--Laplace Urn
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917774685634560 |
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| author | Olesker-Taylor, Sam Schmid, Dominik |
| author_facet | Olesker-Taylor, Sam Schmid, Dominik |
| contents | We analyse the convergence to equilibrium of the Bernoulli--Laplace urn model: initially, one urn contains $k$ red balls and a second $n-k$ blue balls; in each step, a pair of balls is chosen uniform and their locations are switched. Cutoff is known to occur at $\tfrac12 n \log \min\{k, \sqrt n\}$ with window order $n$ whenever $1 \ll k \le \tfrac12 n$. We refine this by determining the limit profile: a function $Φ$ such that \[
d_\mathsf{TV}\bigl( \tfrac12 n \log \min\{k, \sqrt n\} + θn \bigr) \to
Φ(θ) \quad\text{as}\quad
n \to \infty \quad\text{for all}\quad
θ\in \mathbb R. \] Our main technical contribution, of independent interest, approximates a rescaled chain by a diffusion on $\mathbb R$ when $k \gg \sqrt n$, and uses its explicit law as a Gaussian process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit Profile for the Bernoulli--Laplace Urn Olesker-Taylor, Sam Schmid, Dominik Probability Combinatorics 60J10, 60J60, 37A25, 37A30 We analyse the convergence to equilibrium of the Bernoulli--Laplace urn model: initially, one urn contains $k$ red balls and a second $n-k$ blue balls; in each step, a pair of balls is chosen uniform and their locations are switched. Cutoff is known to occur at $\tfrac12 n \log \min\{k, \sqrt n\}$ with window order $n$ whenever $1 \ll k \le \tfrac12 n$. We refine this by determining the limit profile: a function $Φ$ such that \[ d_\mathsf{TV}\bigl( \tfrac12 n \log \min\{k, \sqrt n\} + θn \bigr) \to Φ(θ) \quad\text{as}\quad n \to \infty \quad\text{for all}\quad θ\in \mathbb R. \] Our main technical contribution, of independent interest, approximates a rescaled chain by a diffusion on $\mathbb R$ when $k \gg \sqrt n$, and uses its explicit law as a Gaussian process. |
| title | Limit Profile for the Bernoulli--Laplace Urn |
| topic | Probability Combinatorics 60J10, 60J60, 37A25, 37A30 |
| url | https://arxiv.org/abs/2409.07900 |