Shuffling cards by spatial motion

Persi Diaconis and Soumik Pal

Stochastic Processes and their Applications, 2022, vol. 152, issue C, 149-176

Abstract: 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 Ologm. 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.

Keywords: Markov chains mixing time; Card shuffling; Reflected diffusions; Skorokhod maps; Planar Brownian motion; Stochastic flow of kernels (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2022.06.023

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