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A note on the asymptotics of the free energy of $1+1$ dimensional directed polymers in random environment at high temperature

Makoto Nakashima

TL;DR

This paper removes the concentration-inequality assumption used in prior work to obtain the high-temperature asymptotics of the free energy for the 1+1 dimensional DPRE. By constructing a coarse-grained space-time lattice and an associated oriented percolation framework, the authors establish a robust lower bound that matches the continuum KPZ-based prediction $F(\beta) \sim -\beta^4/6$ as $\beta\to 0$, identifying the limit with the continuum directed polymer free energy $F_{\mathcal Z}(\sqrt{2})=-1/6$. The proof hinges on a Brownian scaling limit connecting discrete DPRE to the continuum polymer, and employs reflection principles, Chernoff bounds, hypercontractivity, and a Garsia-Rodemich-Rumsey type regularity argument to control fluctuations and tube-out contributions. This strengthens the universality perspective by linking discrete models directly to KPZ-related continuum objects without extra concentration hypotheses, with potential extensions to related disordered systems such as pinning models.

Abstract

The author gave the sharp asymptotic behavior of the free energy of $1+1$ dimensional directed polymers in random environment(DPRE) as the inverse temperature $β\to 0$ under the assumption that random environment satisfies a certain concentration inequality in [Nak19], \[\lim_{β\to0}\frac{1}{β^4}F(β)=-\frac{1}{6}. \] In this paper, we obtain the same asymptotics without using the concentration inequality.

A note on the asymptotics of the free energy of $1+1$ dimensional directed polymers in random environment at high temperature

TL;DR

This paper removes the concentration-inequality assumption used in prior work to obtain the high-temperature asymptotics of the free energy for the 1+1 dimensional DPRE. By constructing a coarse-grained space-time lattice and an associated oriented percolation framework, the authors establish a robust lower bound that matches the continuum KPZ-based prediction as , identifying the limit with the continuum directed polymer free energy . The proof hinges on a Brownian scaling limit connecting discrete DPRE to the continuum polymer, and employs reflection principles, Chernoff bounds, hypercontractivity, and a Garsia-Rodemich-Rumsey type regularity argument to control fluctuations and tube-out contributions. This strengthens the universality perspective by linking discrete models directly to KPZ-related continuum objects without extra concentration hypotheses, with potential extensions to related disordered systems such as pinning models.

Abstract

The author gave the sharp asymptotic behavior of the free energy of dimensional directed polymers in random environment(DPRE) as the inverse temperature under the assumption that random environment satisfies a certain concentration inequality in [Nak19], In this paper, we obtain the same asymptotics without using the concentration inequality.

Paper Structure

This paper contains 10 sections, 11 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Model and main result
  3. Outline of the proof
  4. Proof of Theorem \ref{['thm:main']}
  5. Coarse grained argument deduced from DPRE
  6. Coarse grained oriented site percolation
  7. Proof of Lemma \ref{['lem:Berdens']}
  8. Continuum directed polymers from DPRE
  9. Proof of Lemma \ref{['lem:wout']} and Lemma \ref{['lem:wexp']}
  10. Local limit theorem

Key Result

Theorem 1.1

Suppose $d=1$ and eq:normalization is satisfied. Then, we have

Figures (1)

  • Figure 1: Image of SRW trace in tubes: Bold horizontal segments are CG lattice sites. Simple random walk starts from the origin and goes through the site in CG lattice sites without escaping diagonal line areas.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof : Proof of \ref{['eq:lower']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.2
  • Lemma 2.3
  • ...and 13 more