Limiares críticos e distribuição limite para o modelo Bak-Sneppen

Abstract

One of the key problems related to the Bak-Sneppen evolution model is to compute the limit distribution of the tnesses in the stationary regime, as the size of the system tends to in nity. Simulations in [2], [13] and [10] suggest that the one-dimensional limit marginal distribution is uniform on (pc, 1), for some pc 0:667. The article Critical Thresholds and the Limit Distribution in the Bak-Sneppen Model [23] presents a series of relevant results to this conjecture. Our objective will then be study this article, detailing its demonstration in the best possible way. We will de ne three critical thresholds related to avalanche characteristics. We prove that if these critical thresholds are the same and equal to some pc (it has only been proved that two of them are the same) then the limit distribution is the product of uniform distributions on (pc, 1), and moreover pc < 0:75. Our proofs are based on a self-similar graphical representation of the avalanches.