Slowly driven dissipative systems may evolve to a critical state where long
periods of apparent equilibrium are punctuated by intermittent avalanches of
activity. We present a self-organized critical model of punctuated equilibrium
behavior in the context of biological evolution, and solve it in the limit that
the number of independent traits for each species diverges. We derive an exact
equation of motion for the avalanche dynamics from the microscopic rules. In
the continuum limit, avalanches propagate via a diffusion equation with a
nonlocal, history-dependent potential representing memory. This nonlocal
potential gives rise to a non-Gaussian (fat) tail for the subdiffusive
spreading of activity. The probability for the activity to spread beyond a
distance r in time s decays as Ο24ββsβ3/2x1/3exp[β43βx1/3] for x=sr4ββ«1. The potential
represents a hierarchy of time scales that is dynamically generated by the
ultrametric structure of avalanches, which can be quantified in terms of
``backward'' avalanches. In addition, a number of other correlation functions
characterizing the punctuated equilibrium dynamics are determined exactly.Comment: 44 pages, Revtex, (12 ps-figures included