  An effective-medium theory is proposed for random weakly nonlinear dielectric
media. It is based on a new gaussian approximation for the probability
distributions of the electric field in each component of a multi-phase
composite. These distributions are computed to linear order from a
Bruggeman-like self-consistent formula. The resulting effective-medium formula
for the nonlinear medium reduces to Bruggeman's in the linear case. It is exact
up to second order in a weak-disorder expansion, and close to the exact result
in the dilute limit (in particular, it is exact for d=1 and d=infinity. In a
high contrast situation, the noise exponents are kappa=kappa'=0 near the
percolation threshold. Numerical results are provided for different weak
nonlinearities.
