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**Scaling limits of directed polymers in spatial-correlated environment.**
*(English)*
Zbl 1534.60138

Summary: We consider a directed polymer model in dimension \(1 + 1\), where the random walk is attracted to stable law and the environment is independent in time variable and correlated in space variable. We obtain the scaling limit in the intermediate disorder regime for partition function, and show that the rescaled point-to-point partition function of directed polymers converges in the space of continuous functions to the solution of a stochastic heat equation driven by time-white spatial-colored noise. The scaling limit of the polymer transition probability is also established in the path space. The proof of the tightness is based on the gradient estimates for symmetric random walks in the domain of normal attraction of \(\alpha \)-stable law which are established in this paper.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

82D60 | Statistical mechanics of polymers |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60F05 | Central limit and other weak theorems |

### Keywords:

directed polymer; stochastic heat equation; random walk; stable law; spatial-correlated environment
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\textit{Y. Chen} and \textit{F. Gao}, Electron. J. Probab. 28, Paper No. 68, 57 p. (2023; Zbl 1534.60138)

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