Grain Sorting in the One-dimensional Sand Pile Model
Résumé
We study the evolution of a one-dimensional pile, empty at first, which receives a grain in its first stack at each iteration. The final position of grains is singular: grains are sorted according to their parity. They are sorted on trapezoidal areas alternating on both sides of a diagonal line of slope $\sqrt{2}$. This is explained and proved by means of a local study. Each generated pile, encoded in height differences, is the concatenation of four patterns: $22$, $1313$, $0202$, and $11$. The relative length of the first two patterns and the last two patterns converges to $\sqrt{2}$. We make asymptotic expansions and prove that all the lengths of the pile are increasing proportionally to the square root of the number of iterations.