PARTITIONED FRAMES IN DISCRETE BAK SNEPPEN MODELS

In this paper, we wish to present some simpli¯ed cases of discrete Bak- Sneppen models in which explicit computations via Markov chains are possible, hence reaching a better understanding of some rather hidden phenomena of the general case: in particular "avalanches" can be read in terms of mean waiting times and in terms of transitions between structures. The simple models allow us to introduce new frames that do not seem to have been considered in the previous literature, namely the case of partitioned Bak-Sneppen frames, that appear more realistic from the point of view of speed of evolution and do not present a unique criticality level, but a staircase tending towards a ¯nal equilibrium level, cadenced by an increasing sequence of footholds. The introduction summarizes Bak-Sneppen models, starting from the central model due to Bak and Sneppen, and recalls their use in applied sciences. The ¯rst section gives the ge- neral frame of models where locality and globality coexist, the second section shows the simplest case of a matching between locality and globality, that will become exemplar in the most complex frames of Bak-Sneppen processes. The main quantitative theorems are stated and proved in the third section and ¯nally the fourth section presents exam- ples that illustrate the more sophisticated points of our paper and the use (and limits) of experimental results, while the ¯fth section considers real world situations where Bak Sneppen partitioned schemes can be tailored to represent the core of their evolution.