Limit Profile for the Bernoulli--Laplace Urn

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 $\frac{1}{2}n\log \min \{k,\sqrt{n}\}$ with window order $n$ whenever $1\ll k\le \frac{1}{2}n$. We refine this by determining the limit profile: a function $\hat{I}¦$ 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.

19 pages