$d_c=4\mathrm{}$ Is the Upper Critical Dimension for the Bak-Sneppen Model

Numerical results are presented indicating $d_c=4\mathrm{}$ 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 (1998)] that placed $d_c$ as high as $d=8\mathrm{}$. In particular, we find that avalanches are compact for all dimensions $d\le 4$ and are fractal for $d>\mathrm{}4\mathrm{}$. Under those conditions, scaling arguments predict a $d_c=4\mathrm{}$, where hyperscaling relations hold for $d\le 4$. Other properties of avalanches, studied for $1\le d\le 6$, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.