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On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window

By Francesco Caravenna, Rongfeng Sun and Nikos Zygouras


The partition function of the directed polymer model on Z^{2+ 1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^ 2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on R^ 2, extending previous work by Bertini and Cancrini

Topics: QA, QA76
Publisher: 'Springer Science and Business Media LLC'
Year: 2019
DOI identifier: 10.1007/s00220-019-03527-z
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