Shuffling cards by spatial motion

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 $O\left(\log m\right)$. 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.