  We prove the emergence of stable fluctuations for reaction-diffusion in
random environment with Weibull tails. This completes our work around the
quenched to annealed transition phenomenon in this context of reaction
diffusion. In [9], we had already considered the model treated here and had
studied fully the regimes where the law of large numbers is satisfied and where
the fluctuations are Gaussian, but we had left open the regime of stable
fluctuations. Our work is based on a spectral approach centered on the
classical theory of rank-one perturbations. It illustrates the gradual
emergence of the role of the higher peaks of the environments. This approach
also allows us to give the delicate exact asymptotics of the normalizing
constants needed in the stable limit law.
