Stochastic Box-Ball System

David Keating
University of Wisconsin-Madison

The box-ball system is a deterministic discrete-time dynamical system introduced by Takahashi and Satsuma. At each natural number suppose there is a box that can either hold a single ball or be empty. A ball configuration is a sequence of 0's and 1's, 1 denoting a box contains a ball and 0 denoting an empty box. To evolve the ball configuration one step in time, a `carrier' sweeps across the from left to right starting at zero. Whenever the carrier encounters a ball, it picks the ball up, leaving the corresponding box empty. Whenever it reaches an empty box while holding at least one ball, it fills the box by dropping one of its carried balls.

In this talk, we introduce a stochastic version of the box-ball dynamics. Rather than the carrier deterministically picking up each ball during a sweep, there is now a failure probability $\epsilon$ such that, independently for each ball, the carrier fails to pick up the ball with probability $\epsilon$. The object of interest will not be the positions of the balls themselves, but instead the spacing between them, which we call the inter-distance configuration. Our main result is that the inter-distance configuration, after suitable rescaling, converges to semi-martingale reflecting Brownian motion.

Presentation (PDF File)

Back to Workshop IV: Vertex Models: Algebraic and Probabilistic Aspects of Universality