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High moments of 2d directed polymers up to quasi-criticality

Clément Cosco, Shuta Nakajima

TL;DR

This work analyzes two-dimensional directed polymers in random environments under subcritical and quasi-critical scaling, deriving upper bounds for the q-th moments of the partition function up to a diverging threshold q_N. The authors develop a diagrammatic expansion and a refined induction scheme that separates pairwise from higher-order intersections, enabling Gaussian-type moment bounds to hold beyond previously known thresholds and for general weight distributions. They also prove Gaussian-rate large deviation results for log W_N in both regimes and provide a moment-formula framework that extends to non-Gaussian weights. The results sharpen our understanding of high-moment behavior and its connections to extreme-value statistics and log-correlated structures in 2D random polymers, with potential implications for critical and near-critical phenomena. The methodology blends diagrammatic expansions, truncations, long-jump inductions, and careful probabilistic-combinatorial control to bridge subcritical, quasi-critical, and higher-q regimes.

Abstract

We consider two-dimensional directed polymers in random environment in the sub-critical regime and in the quasi-critical regime introduced recently by Caravenna, Cottini and Rossi, arXiv:2307.02453v1. For $q\leq q_N$ with $q_N\to\infty$ diverging at a suitable rate with the size of the system, we obtain upper bound estimates on the $q$-moment of the partition function for general environments. In the sub-critical regime, our results improve the $q_N$-threshold obtained for Gaussian environment in Cosco, Zeitouni, Comm. Math. Phys (2023). As a corollary, we derive large deviation estimates with a Gaussian rate function.

High moments of 2d directed polymers up to quasi-criticality

TL;DR

This work analyzes two-dimensional directed polymers in random environments under subcritical and quasi-critical scaling, deriving upper bounds for the q-th moments of the partition function up to a diverging threshold q_N. The authors develop a diagrammatic expansion and a refined induction scheme that separates pairwise from higher-order intersections, enabling Gaussian-type moment bounds to hold beyond previously known thresholds and for general weight distributions. They also prove Gaussian-rate large deviation results for log W_N in both regimes and provide a moment-formula framework that extends to non-Gaussian weights. The results sharpen our understanding of high-moment behavior and its connections to extreme-value statistics and log-correlated structures in 2D random polymers, with potential implications for critical and near-critical phenomena. The methodology blends diagrammatic expansions, truncations, long-jump inductions, and careful probabilistic-combinatorial control to bridge subcritical, quasi-critical, and higher-q regimes.

Abstract

We consider two-dimensional directed polymers in random environment in the sub-critical regime and in the quasi-critical regime introduced recently by Caravenna, Cottini and Rossi, arXiv:2307.02453v1. For with diverging at a suitable rate with the size of the system, we obtain upper bound estimates on the -moment of the partition function for general environments. In the sub-critical regime, our results improve the -threshold obtained for Gaussian environment in Cosco, Zeitouni, Comm. Math. Phys (2023). As a corollary, we derive large deviation estimates with a Gaussian rate function.

Paper Structure

This paper contains 44 sections, 46 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Results
  4. Moment bounds
  5. Estimates for varying starting and ending times
  6. Large and moderate deviations estimates.
  7. Moment formula
  8. Comparison with related works
  9. Notation and Gaussian weights.
  10. Description of the proof
  11. Structure of the paper.
  12. Expansion into successive disjoint pairings.
  13. Truncated moment estimates
  14. Truncation
  15. The general $p_0$ case
  16. ...and 29 more sections

Key Result

Theorem 2.2

Assume eq:sub-criticality. Let $\alpha \in (0,1)$. There exists $C_\alpha > 0$ such that for all $q\in \mathbb{N}_{\geq 3}$ satisfying we have with $\lambda^2(\hat{\beta}):=\log (\frac{1}{1-\hat{\beta}^2})$, where $o_N(1)\to 0$ as $N\to\infty$, uniformly over all $q\in \mathbb{N}_{\geq 3}$ satisfying eq:q-conditionSubClassic.

Theorems & Definitions (105)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5: finite $q$
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 95 more