### 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 Ben Arous et al (Transition asymptotics for reaction-diffusion in random media. Probability and mathematical physics, American Mathematical Society, Providence, RI, pp 1–40, 2007, [8]), 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.

Original language | English (US) |
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Title of host publication | Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016 |

Editors | Stefano Olla, Peter Friz, Wolfgang König, Chiranjib Mukherjee |

Publisher | Springer New York LLC |

Pages | 123-171 |

Number of pages | 49 |

ISBN (Print) | 9783030153373 |

DOIs | |

State | Published - Jan 1 2019 |

Event | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 - Berlin, Germany Duration: Aug 15 2016 → Aug 19 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 283 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 |
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Country | Germany |

City | Berlin |

Period | 8/15/16 → 8/19/16 |

### Fingerprint

### Keywords

- Principal eigenvalue
- Random walk
- Rank one perturbation theory
- Stable distributions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016*(pp. 123-171). (Springer Proceedings in Mathematics and Statistics; Vol. 283). Springer New York LLC. https://doi.org/10.1007/978-3-030-15338-0_5