### Abstract

The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel [Ann. Probab. 42 (2014) 1212-1256]. It was proved that at inverse temperature βn -γ with γ = 1/4 the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of γ.

Original language | English (US) |
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Pages (from-to) | 4006-4048 |

Number of pages | 43 |

Journal | Annals of Probability |

Volume | 44 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2016 |

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### Keywords

- Directed polymer
- Heavy tail
- Phase transition
- Scaling limits

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*44*(6), 4006-4048. https://doi.org/10.1214/15-AOP1067