  Many systems across the sciences evolve through a combination of
multiplicative growth and diffusive transport. In the presence of disorder,
these systems tend to form localized structures which alternate between long
periods of relative stasis and short bursts of activity. This behaviour, known
as intermittency in physics and punctuated equilibrium in evolutionary theory,
is difficult to forecast; in particular there is no general principle to locate
the regions where the system will settle, how long it will stay there, or where
it will jump next. Here I introduce a predictive theory of linear intermittency
that closes these gaps. I show that any positive linear system can be mapped
onto a generalization of the "maximal entropy random walk", a Markov process on
graphs with non-local transition rates. This construction reveals the
localization islands as local minima of an effective potential, and
intermittent jumps as barrier crossings in that potential. My results unify the
concepts of intermittency in linear systems and Markovian metastability, and
provide a generally applicable method to reduce, and predict, the dynamics of
disordered linear systems. Applications span physics, evolutionary dynamics and
epidemiology.
