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On the Moments of the \((2+1)\)-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

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Abstract

The partition function of the directed polymer model on \({\mathbb {Z}}^{2+1}\) 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 \({\mathbb {R}}^{2}\), 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 \({\mathbb {R}}^2\), extending previous work by Bertini and Cancrini (J Phys A Math Gen 31:615, 1998).

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Notes

  1. Note that coordinates \((n^\alpha _i,x^\alpha _i)\) with the same label \(\alpha \in \{a,b,c\}\) are distinct, by \(n^\alpha _i < n^\alpha _{i+1}\), see (1.39), hence more than triple matchings cannot occur.

  2. \(S^{(N)}\) should not be confused with the random walk S in the definition of the directed polymer model.

  3. For simplicity, in relations (5.40) we have omitted the “periodicity correction” \(2 \, \mathbb {1}_{\{(n,x) \in {\mathbb {Z}}^3_{\mathrm {even}}\}}\), see (1.8) and (2.22), because this disappears upon summation.

  4. The precise constant \(4 \pi \) in (8.2) is the one relevant for us, but any other positive constant would do.

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Acknowledgements

F.C. is supported by the PRIN Grant 20155PAWZB “Large Scale Random Structures”. R.S. is supported by NUS Grant R-146-000-253-114. N.Z. is supported by EPRSC through Grant EP/R024456/1.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 55, 20125, Milan, Italy

    Francesco Caravenna

  2. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Rongfeng Sun

  3. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    Nikos Zygouras

Authors
  1. Francesco Caravenna
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  2. Rongfeng Sun
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  3. Nikos Zygouras
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Correspondence to Rongfeng Sun.

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Communicated by M. Hairer

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Caravenna, F., Sun, R. & Zygouras, N. On the Moments of the \((2+1)\)-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window. Commun. Math. Phys. 372, 385–440 (2019). https://doi.org/10.1007/s00220-019-03527-z

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