2017/08/27 by Persi Diaconis, Soumik Pal, Diaconis, Persi +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60J10 #60J60 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1708.08147
openalex publication_date 2017/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a model of card shuffling where a pack of cards, spread as points on a square table, are repeatedly gathered locally at random spots and then spread towards a random direction. A shuffling of the cards is then obtained by arranging the cards by their increasing x-coordinate values. When there are m cards on the table we show that this random ordering gets mixed in time O(log m). Explicit constants are evaluated in a diffusion limit when the position of m cards evolves as an interesting 2m-dimensional non-reversible reflected jump diffusion in time. Our main technique involves the use of multidimensional Skorokhod maps for double reflections in [0,1]2 in taking the discrete to continuous limit. The limiting computations are then based on the planar Brownian motion and properties of Bessel processes.