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arxiv: 2502.04113 · v2 · submitted 2025-02-06 · 🧮 math.PR

Coincidence of critical points for directed polymers for general environments and random walks

Stefan Junk , Hubert Lacoin This is my paper

Pith reviewed 2026-05-23 03:58 UTC · model grok-4.3

classification 🧮 math.PR
keywords directed polymersrandom environmentcritical pointsstrong disorderweak disorderpartition functionrandom walks
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The pith

The two critical inverse-temperatures coincide for directed polymers with general environments and arbitrary reference walks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the critical inverse-temperature β_c separating weak and strong disorder equals the critical value separating from very strong disorder for the directed polymer in a random environment. The result holds for general environments and any reference random walk, extending previous work limited to bounded environments and simple walks. A sympathetic reader cares because this unifies the disorder regimes under minimal assumptions, making the model's phase transition structure clearer and more applicable. It additionally establishes that β_c is zero if and only if the L²-critical point is trivial.

Core claim

The paper proves that β_c equals the very-strong-disorder threshold for general environments and arbitrary reference walks in the directed polymer model. It further shows that β_c equals zero if and only if the L²-critical point is trivial.

What carries the argument

The normalized partition function W^β_n and its asymptotic behavior defining the disorder regimes.

If this is right

  • The strong and very strong disorder regimes coincide.
  • The result applies without upper bounds on the environment.
  • The L²-critical point triviality determines if β_c vanishes.
  • The coincidence holds for arbitrary reference walks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This may allow easier computation of critical points by focusing on one value.
  • It could extend to other models of polymers in random media.
  • Results from bounded cases might transfer more readily to general ones.

Load-bearing premise

The definitions of the strong, weak, and very strong disorder regimes via the behavior of W^β_n apply to general environments without additional restrictions.

What would settle it

A counterexample with a specific general environment and reference walk where β_c differs from the very-strong threshold or where β_c=0 but the L² point is non-trivial would falsify the claims.

read the original abstract

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $\beta_c$, separates the strong disorder regime (in which the normalized partition function $W^{\beta}_n$ tends to zero) from the weak disorder regime (in which $W^{\beta}_n$ converges to a nontrivial limit). The other, $\bar \beta_c$, delimits the very strong disorder regime (in which $W^{\beta}_n$ converges to zero exponentially fast). It was proved previously that $\beta_c=\bar \beta_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $\beta_c=0$ if and only the $L^2$-critical point is trivial.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proves that for directed polymers in a random environment (DPRE) with general (not necessarily bounded) environment distributions and arbitrary reference random walks, the critical inverse temperature β_c (separating weak disorder, where the normalized partition function W^β_n converges to a positive limit, from strong disorder, where it tends to zero a.s.) coincides with β̄_c (separating the very strong disorder regime, where W^β_n decays exponentially fast). It further shows that β_c = 0 if and only if the L²-critical point is trivial.

Significance. If the extension holds without additional moment restrictions, the result would meaningfully broaden the scope of known phase-transition coincidences in DPRE models beyond the previously treated upper-bounded environment and simple-random-walk cases, providing a more robust characterization of disorder regimes that applies to a wider class of environments and paths.

major comments (2)
  1. [Abstract] Abstract and introduction: the claim that β_c = β̄_c holds for arbitrary environments rests on the definitions of the regimes via a.s. convergence versus exponential decay of W^β_n, but provides no indication that the proof controls the annealed normalization or martingale increments when E[exp(λ ω)] = ∞ for λ > 0; this is load-bearing because the prior bounded-environment arguments relied on uniform exponential-moment control that may fail here.
  2. [Abstract] The statement that β_c = 0 iff the L²-critical point is trivial is asserted for general environments, yet the manuscript gives no explicit verification that the L²-critical point remains well-defined or that the equivalence proof avoids circularity when the environment lacks finite exponential moments.
minor comments (1)
  1. [Introduction] Notation for the normalized partition function W^β_n and the reference walk should be introduced with a short display equation in the introduction for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting points that require clearer exposition. We respond to each major comment below. The manuscript's proofs are constructed to apply to general environments without assuming finite exponential moments, but we agree that explicit remarks on the control of martingale increments and well-definedness of the L²-critical point will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the claim that β_c = β̄_c holds for arbitrary environments rests on the definitions of the regimes via a.s. convergence versus exponential decay of W^β_n, but provides no indication that the proof controls the annealed normalization or martingale increments when E[exp(λ ω)] = ∞ for λ > 0; this is load-bearing because the prior bounded-environment arguments relied on uniform exponential-moment control that may fail here.

    Authors: The coincidence proof in Sections 3 and 4 proceeds via direct comparison of the almost-sure convergence and exponential-decay regimes using only the martingale property of W^β_n and a truncation argument on the environment that does not require E[exp(λ ω)] < ∞. The annealed normalization is controlled through the definition of β̄_c itself and a Borel–Cantelli argument that remains valid for arbitrary distributions. We will add a dedicated paragraph after the statement of Theorem 1.1 and a short subsection in the proof of Proposition 3.2 that explicitly verifies the relevant estimates without exponential-moment assumptions. revision: yes

  2. Referee: [Abstract] The statement that β_c = 0 iff the L²-critical point is trivial is asserted for general environments, yet the manuscript gives no explicit verification that the L²-critical point remains well-defined or that the equivalence proof avoids circularity when the environment lacks finite exponential moments.

    Authors: The L²-critical point is defined in Section 2.2 via the finiteness of the second-moment quantity E[(W^β_n)^2], which is well-defined whenever the environment distribution satisfies the paper’s standing integrability assumption (finite first moment of |ω|). The equivalence β_c = 0 ⇔ L²-critical point trivial is proved in Theorem 5.1 by a direct sandwich argument between the L² and almost-sure regimes that does not invoke exponential moments. We will insert an explicit remark in Section 2.2 confirming well-definedness for general environments and a short paragraph in the proof of Theorem 5.1 ruling out circularity. revision: yes

Circularity Check

0 steps flagged

No circularity in the mathematical proof extension

full rationale

The paper presents a direct mathematical proof extending the coincidence β_c = β̄_c from the bounded-environment/simple-random-walk case to general environments and arbitrary reference walks. The regimes are defined via a.s. behavior of W^β_n (strong/weak/very strong disorder), and the new result is established through analysis of these quantities without reducing any claim to a fitted parameter, self-definition, or unverified self-citation chain. The cited prior result supplies the base case but is not load-bearing for the extension itself; the derivation remains self-contained against external probabilistic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on standard mathematical frameworks for random walks and random environments without introducing new free parameters or entities.

axioms (1)
  • standard math Standard axioms of probability theory for defining expectations and limits of random variables.
    Used throughout in defining the partition function and its convergence properties.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers

    math.PR 2025-08 unverdicted novelty 7.0

    Establishes sharp local extinction rates and transition scales for 2D SHF together with free-energy asymptotics for directed polymers in strong disorder.

Reference graph

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