# Stable limit laws and structure of the scaling function for reaction-diffusion in random environment

## TL;DR

This paper establishes the presence of stable fluctuations in reaction-diffusion systems within random environments exhibiting Weibull tails, completing the understanding of fluctuation regimes and the transition from Gaussian to stable laws.

## Contribution

It introduces a spectral approach to characterize stable fluctuations and provides precise asymptotics for normalization constants in reaction-diffusion models with Weibull tails.

## Key findings

- Stable fluctuations are proven for reaction-diffusion in Weibull tail environments.
- The spectral approach reveals the influence of environment peaks on fluctuations.
- Exact asymptotics for normalization constants in stable limit laws are derived.

## Abstract

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.