Stratifications of cellular patterns: Hysteresis and convergence

A foam is a space-filling cellular pattern, that can be decomposed into successive layers or strata. Each layer contains all cells at the same topological distance to an origin (cell, cluster of cells, or basal layer). The disorder of the underlying structure imposes a characteristic roughening of the layers. In this paper, stratifications are described as the results a deterministic "invasion" process started from different origins in the same, given foam. We compare different stratifications of the same foam. Our main results are 1) hysteresis and 2) convergence in the sequence of layers. 1) If the progression direction is reversed, the layers in the up and down sequences differ (irreversibility of the invasion process); nevertheless, going back up, the layers return exactly to the top profile. This hysteresis phenomenon is established rigorously from elementary properties of graphs and processes. 2) Layer sequences based on different origins (e.g. different starting cells) converge, in cylindrical geometry. Jogs in layers may be represented as pairs of opposite dislocations, that move erratically because the underlying structure is disordered, and end up annihilating when colliding. Convergence is demonstrated and quantified by numerical simulations on a two dimensional columnar model.