  We propose a new mechanism for generating power laws. Starting from a random
walk, we first outline a simple derivation of the Fokker-Planck equation. By
analogy, starting from a certain Markov chain, we derive a master equation for
power laws that describes how the number of cascades changes over time
(cascades are consecutive transitions that end when the initial state is
reached). The partial differential equation has a closed form solution which
gives an explicit dependence of the number of cascades on their size and on
time. Furthermore, the power law solution has a natural cut-off, a feature
often seen in empirical data. This is due to the finite size a cascade can have
in a finite time horizon. The derivation of the equation provides a
justification for an exponent equal to 2, which agrees well with several
empirical distributions, including Richardson's law on the size and frequency
of deadly conflicts. Nevertheless, the equation can be solved for any exponent
value. In addition, we propose an urn model where the number of consecutive
ball extractions follows a power law. In all cases, the power law is manifest
over the entire range of cascade sizes, as shown through log-log plots in the
frequency and rank distributions.
