Grain Sorting in the One-dimensional Sand Pile Model

We study the evolution of a one-dimensional pile, empty at rst, which receives a grain in its rst stack at each iteration. The nal 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 p 2. This is explained and proved by means of a local study. Each generated pile, encoded in height diierences, is the concatenation of four patterns: 22, 1313, 0202, and 11. The relative length of the rst two patterns and the last two patterns converges to p 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.