  Front propagation described by Huygens' principle is a fundamental mechanism
of spatial spreading of a property or an effect, occurring in optics,
acoustics, ecology and combustion. If the local front speed varies randomly due
to inhomogeneity or motion of the medium (as in turbulent premixed combustion),
then the front wrinkles and its overall passage rate (turbulent burning
velocity) increases. The calculation of this speedup is subtle because it
involves the minimum-time propagation trajectory. Here we show mathematically
that for a medium with weak isotropic random fluctuations, under mild
conditions on its spatial structure, the speedup scales with the 4/3 power of
the fluctuation amplitude. This result, which verifies a previous conjecture
while clarifying its scope, is obtained by reducing the propagation problem to
the inviscid Burgers equation with white-in-time forcing. Consequently,
field-theoretic analyses of the Burgers equation have significant implications
for fronts in random media, even beyond the weak-fluctuation limit.
