Limit Profile for the Bernoulli--Laplace Urn

Sam Olesker-Taylor, Dominik Schmid
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Abstract

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 $\Phi$ such that \[ d_\mathsf{TV}\bigl( \tfrac12 n \log \min\{k, \sqrt n\} + \theta n \bigr) \to \Phi(\theta) \quad\text{as}\quad n \to \infty \quad\text{for all}\quad \theta \in \mathbb R. \] 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.
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伯努利-拉普拉斯瓮的极限轮廓
我们分析了伯努利--拉普拉斯瓮模型向均衡收敛的过程:最初,一个瓮包含 $k$ 红球,第二个瓮包含 $n-k$ 蓝球;在每一步中,均匀地选择一对球,并切换它们的位置。众所周知,当 1 \ll k \le \tfrac12 n$ 时,截止点会出现在 $\tfrac12 n \log \min\{k, \sqrt n\}$,窗口阶数为 $n$。我们通过确定极限轮廓来完善这一点:a function $\Phi$ such that \[ d_\mathsf{TV}\bigl( \tfrac12 n \log \min\{k, \sqrt n\} + \theta n \bigr) \to \Phi(\theta) \quad\text{as}\quad n \to \infty \quad\text{for all}\quad \theta \in \mathbb R.\]我们的主要技术贡献是,当 $k\gg \sqrt n$ 时,用 $\mathbb R$ 上的扩散来近似一个重标度链,并将其显式规律作为一个高斯过程。
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