d_c=4 is the upper critical dimension for the Bak-Sneppen model

TL;DR

This paper provides numerical evidence that the upper critical dimension for the Bak-Sneppen model is 4, confirming some theoretical predictions and contradicting recent claims of a higher critical dimension.

Contribution

The study establishes d_c=4 as the upper critical dimension for the Bak-Sneppen model through improved numerical simulations and analysis of avalanche properties across multiple dimensions.

Findings

Avalanches are compact for d<=4 and fractal for d>4.

Scaling relations and avalanche properties support d_c=4.

Numerical algorithm based on branching process enhances analysis accuracy.

Abstract

Numerical results are presented indicating d_c=4 as the upper critical dimension for the Bak-Sneppen evolution model. This finding agrees with previous theoretical arguments, but contradicts a recent Letter [Phys. Rev. Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we find that avalanches are compact for all dimensions d<=4, and are fractal for d>4. Under those conditions, scaling arguments predict a d_c=4, where hyperscaling relations hold for d<=4. Other properties of avalanches, studied for 1<=d<=6, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.