# Dead Leaves Simulation

###### LEFT-HAND SIDE:

**Imperfect dead leaves simulation** (forwards or conventional construction). The leaves fall down on the ground, as you view them from above. In the limit at time infinity there is a statistical equilibrium, which is by definition the dead leaves tessellation (if we record only the visible parts of leaf boundaries) or coloured mosaic (if we take account of the visible textures). However at any fixed time the statistical properties are biassed by the initial configuration (as is easy to see: there is a small probability that some patch is left vacant by the falling leaves, which could have been filled in an entirely arbitrary manner by the original configuration!).

###### RIGHT-HAND SIDE:

**Perfect dead leaves simulation** (backwards construction). Change your point of view! Consider a small animal looking up from a hole in the ground. The leaves fall down to form a pattern which is statistically very similar to the pattern viewed from above. (**Prove** this by noting that at any fixed time one could simply re-order the leaves, last becoming first, first becoming last, and so move between a forwards and a backwards construction.) However there comes a time when the early leaves cover the hole completely. After then the tessellation/coloured mosaic ceases to change (in our example this happens late on, at about leaf number 52). From this random time onwards, the simulation is *virtually* a simulation in statistical equilibrium, and is therefore a perfect simulation of the dead leaves tessellation!

###### A NATURAL QUESTION:

Suppose we run the forwards simulation till the time when the leaves completely cover the region. Is it then a perfect sample of the dead leaves tessellation? See Kendall and Thönnes for a simple example showing why the answer is **NO!**