Limit Profile for the Bernoulli--Laplace Urn

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 $\mathrm{\Phi }$ such that $$d_{\mathsf{T}\mathsf{V}}(\frac{1}{2}n\log \min \{k,\sqrt{n}\}+\theta n)\to \mathrm{\Phi }(\theta )\text{as}n\to \mathrm{\infty }\text{for all}\theta \in \mathbb{R}\mathrm{.}$$ 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.