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Quasi-critical fluctuations for 2d directed polymers

Francesco Caravenna, Francesca Cottini, Maurizia Rossi

TL;DR

This work analyzes the 2d directed polymer in random environment in a quasi-critical regime that bridges sub-critical and critical behavior. The authors prove Edwards-Wilkinson fluctuations for diffusively rescaled, averaged partition functions, yielding Gaussian limits with a kernel-defined variance and linking to the critical 2d Stochastic Heat Flow. The proof navigates the failure of hypercontractivity in this regime by developing new high-moment bounds for a polynomial chaos expansion and applying a Lyapunov-based CLT through a block decomposition. A detailed, non-asymptotic moment framework bounds boundary and bulk contributions, enabling precise variance calculations and the Lyapunov condition. The results deepen the connection between directed polymers, stochastic PDEs, and Gaussian fluctuation regimes, with potential extensions to SHF, mollified KPZ, and higher-dimensional settings.

Abstract

We study the 2d directed polymer in random environment in a novel *quasi-critical regime*, which interpolates between the much studied sub-critical and critical regimes. We prove Edwards-Wilkinson fluctuations throughout the quasi-critical regime, showing that the diffusively rescaled partition functions are asymptotically Gaussian. We deduce a corresponding result for the critical 2d Stochastic Heat Flow. A key challenge is the lack of hypercontractivity, which we overcome deriving new moment estimates.

Quasi-critical fluctuations for 2d directed polymers

TL;DR

This work analyzes the 2d directed polymer in random environment in a quasi-critical regime that bridges sub-critical and critical behavior. The authors prove Edwards-Wilkinson fluctuations for diffusively rescaled, averaged partition functions, yielding Gaussian limits with a kernel-defined variance and linking to the critical 2d Stochastic Heat Flow. The proof navigates the failure of hypercontractivity in this regime by developing new high-moment bounds for a polynomial chaos expansion and applying a Lyapunov-based CLT through a block decomposition. A detailed, non-asymptotic moment framework bounds boundary and bulk contributions, enabling precise variance calculations and the Lyapunov condition. The results deepen the connection between directed polymers, stochastic PDEs, and Gaussian fluctuation regimes, with potential extensions to SHF, mollified KPZ, and higher-dimensional settings.

Abstract

We study the 2d directed polymer in random environment in a novel *quasi-critical regime*, which interpolates between the much studied sub-critical and critical regimes. We prove Edwards-Wilkinson fluctuations throughout the quasi-critical regime, showing that the diffusively rescaled partition functions are asymptotically Gaussian. We deduce a corresponding result for the critical 2d Stochastic Heat Flow. A key challenge is the lack of hypercontractivity, which we overcome deriving new moment estimates.
Paper Structure (39 sections, 18 theorems, 211 equations)

This paper contains 39 sections, 18 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. The phase transition
  3. Main result
  4. Strategy of the proof and comparison with the literature
  5. Relevant context and future perspectives
  6. Organization of the paper
  7. Proof of Theorems \ref{['th:EWquasicrit']} and \ref{['th:EW-SHF']}
  8. First step
  9. Second step
  10. Proof of Theorem \ref{['th:EW-SHF']}
  11. Second moment bounds: proof of Proposition \ref{['prop:approximationZbysum']}
  12. Polynomial chaos expansion
  13. Proof of Proposition \ref{['prop:approximationZbysum']}
  14. General moment bounds
  15. Moment expansion
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $Z_{N,\beta}^\omega(\varphi)$ denote the diffusively rescaled and averaged partition function of the 2d directed polymer model, see eq:Z and eq:ZNav, for disorder variables $\omega$ which satisfy eq:omega. Then, for $(\beta_N)_{N\in\mathbb{N}}$ in the quasi-critical regime, see eq:sigma and eq:q where the limiting variance is given by

Theorems & Definitions (36)

  • Theorem 1.1: Quasi-critical Edwards-Wilkinson fluctuations
  • Theorem 1.2: Edwards-Wilkinson fluctuations for the SHF
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1: $L^2$ approximation
  • Remark 2.2
  • Proposition 2.3: Fourth moment bound
  • Remark 3.1
  • Lemma 3.2: Quasi-critical variance
  • ...and 26 more