Limit Profile for the Bernoulli--Laplace Urn

2024/09/12 by Sam Olesker-Taylor, Dominik Schmid, Olesker-Taylor, Sam +1 · 2 citations
Engineering · Mathematics · #37A25 #37A30 #60J10 #60J60 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR) #Sports Dynamics and Biomechanics

paper · pdf · doi:10.48550/arxiv.2409.07900

openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, √ n\ with window order n whenever 1 ≪ k ≤ \tfrac12 n. We refine this by determining the limit profile: a function Φ such that d_TV( \tfrac12 n log min\k, √ n\ + θn ) → Φ(θ) \quadas n → ∞ \quadfor all θ∈ \mathbb R. Our main technical contribution, of independent interest, approximates a rescaled chain by a diffusion on \mathbb R when k ≫ √ n, and uses its explicit law as a Gaussian process.

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• Limit profiles of ASEP
• Every cutoff profile is possible

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