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arxiv: 2508.02478 · v3 · submitted 2025-08-04 · 🧮 math.PR

Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers

Quentin Berger , Francesco Caravenna , Nicola Turchi This is my paper

Pith reviewed 2026-05-19 01:02 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic heat flowdirected polymersstrong disorderlocal extinctionmeasure-valued processfree energycoarse grainingchange of measure
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The pith

The critical 2D stochastic heat flow undergoes local extinction at a sharp rate in the strong-disorder regime, with an identified spatial scale for the transition from vanishing to diverging mass.

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

This paper studies the critical two-dimensional stochastic heat flow in the combined large-time and strong-disorder setting. It proves that the measure-valued distribution collapses locally to zero at an explicit rate. The analysis also locates the precise spatial scale at which total mass switches from vanishing to diverging and at which local extinction is replaced by averaged behavior. Parallel statements are obtained for the partition functions of two-dimensional directed polymers, including exact free-energy asymptotics. The results clarify the large-scale fluctuations of space-time discretizations of the stochastic heat equation, which remain superdiffusive for any fixed positive disorder strength.

Core claim

In the large-time and strong-disorder regimes the critical 2D Stochastic Heat Flow exhibits local extinction at a sharp rate, so that its distribution collapses to zero according to an identified decay law. The spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to averaged behavior, is determined explicitly. The same conclusions hold for the partition functions of 2D directed polymers, where they yield precise free-energy estimates. All statements are obtained from a unified framework of change-of-measure and coarse-graining arguments.

What carries the argument

Change-of-measure and coarse-graining arguments that control the strong-disorder regime uniformly for the universal measure-valued critical 2D Stochastic Heat Flow.

If this is right

  • Partition functions of 2D directed polymers admit precise free-energy estimates in the strong-disorder regime.
  • Space-time discretizations of the 2D stochastic heat equation exhibit fluctuations on a superdiffusive scale for any fixed supercritical disorder strength.
  • The transition spatial scale separates regimes of vanishing mass from regimes of diverging mass.
  • Local extinction occurs at an explicitly identifiable rate that governs the collapse of the measure-valued distribution.

Where Pith is reading between the lines

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

  • The identified scales may be used to design efficient numerical schemes that capture the transition between extinction and averaged regimes.
  • Similar change-of-measure techniques could be applied to other two-dimensional disordered systems whose scaling limits involve measure-valued processes.
  • The superdiffusive fluctuation result suggests that macroscopic observables in related polymer models remain sensitive to microscopic disorder even at large scales.

Load-bearing premise

The critical 2D stochastic heat flow exists as a well-defined universal measure-valued process and the disorder distribution permits uniform change-of-measure and coarse-graining controls in the strong-disorder regime.

What would settle it

A direct numerical check of the discretized stochastic heat equation showing that the local mass at the predicted spatial scale and time horizon decays at a rate different from the one derived in the paper.

Figures

Figures reproduced from arXiv: 2508.02478 by Francesco Caravenna, Nicola Turchi, Quentin Berger.

Figure 1
Figure 1. Figure 1: Illustration of an “interaction diagram”. Pairwise interactions are regrouped in stretches of “same-type” interaction labeled by Cr, because of the term q(Cr ⊔ Crˆ)q((Cr ⊔ Crˇ)). A labeled diagram then corresponds to a collections of stretches with a label dp, a length (cardinality) kp and ordered starting and ending points (ap, xp), bp, yp. In the above diagram, there are ℓ = 5 stretches. We can then inte… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the interpretation of the formula (5.3) for Jℓ in terms of alternating stretches of two independent (and identically distributed) renewals τ, τ ′ . The first point is chosen uniformly in J1, N˜K and they then have inter-arrival distribution K(m) defined above. The stretches alternate between τ and τ ′ and have starting and ending point denoted by ap, bp, in reference to the interaction diag… view at source ↗
read the original abstract

The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process that provides a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the large-time and strong-disorder regimes, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. Our proof provides a unified framework of change of measure and coarse-graining arguments. These results offer new insights into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $\beta$, including the case where $\beta > 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.

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

1 major / 2 minor

Summary. The manuscript investigates the critical 2D Stochastic Heat Flow (SHF) in large-time and strong-disorder regimes. It proves a sharp form of local extinction by identifying the rate at which the distribution collapses to zero. It also identifies the spatial scale governing the transition from vanishing mass to diverging mass and from extinction to averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. The proofs rely on a unified framework of change of measure and coarse-graining arguments. The work offers insights into the 2D stochastic heat equation regularized via space-time discretization for any supercritical disorder strength β, including fixed β > 0.

Significance. If the central claims hold, the paper advances the field by providing sharp quantitative results on local extinction and scaling transitions for the critical 2D SHF and directed polymers in the strong-disorder regime. The explicit controls that remain uniform down to fixed supercritical β, along with the derivation of tail estimates and renormalization constants, strengthen the contribution relative to prior qualitative analyses.

major comments (1)
  1. The uniformity of coarse-graining controls for fixed β > 0 is load-bearing for the main extinction rate claim, but the argument in the change-of-measure step requires explicit verification that the tail estimates do not deteriorate as β is held fixed rather than scaled.
minor comments (2)
  1. Notation for the spatial scale in the transition regime should be defined earlier to improve readability of the main theorems.
  2. A brief comparison table of the extinction rates for SHF versus directed polymers would clarify the correspondence between the two settings.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The manuscript aims to provide sharp quantitative results on local extinction and transition scales for the critical 2D SHF and directed polymers, with controls uniform in the fixed supercritical regime. We address the major comment below.

read point-by-point responses

  1. Referee: The uniformity of coarse-graining controls for fixed β > 0 is load-bearing for the main extinction rate claim, but the argument in the change-of-measure step requires explicit verification that the tail estimates do not deteriorate as β is held fixed rather than scaled.

    Authors: We thank the referee for this observation. The change-of-measure construction in Section 3 employs a fixed tilt whose strength is proportional to the given β > 0; the resulting exponential moments of the Radon-Nikodym derivative are controlled by the renormalization constant C_β, which remains bounded and continuous for β in any compact interval [β_0, ∞) with β_0 > 0. The tail bounds in Lemma 3.2 and Proposition 3.4 are then derived from Burkholder-Davis-Gundy estimates whose constants depend only on the fixed β through these moments and on the coarse-graining length scale, which is held fixed once β is fixed. Consequently the estimates do not deteriorate. Nevertheless, to make the uniformity fully explicit we will add a short remark immediately after the statement of Proposition 3.4 that records the β-independence of the constants appearing in the tail probabilities (3.12) and (3.15) when β is held fixed. This is a clarification rather than a new argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The manuscript establishes its main results on local extinction rates and spatial scales for the critical 2D SHF and directed-polymer partition functions through an explicit change-of-measure plus coarse-graining argument that remains uniform in the strong-disorder regime. All tail estimates and renormalization constants are derived directly from the model assumptions rather than fitted to the target quantities or imported via self-citation chains. The central claims therefore reduce to the stated controls on the disorder distribution and the well-definedness of the universal measure-valued process, without any step that equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the prior construction of the critical 2D SHF as a universal object and on standard properties of Gaussian disorder; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The critical 2D Stochastic Heat Flow exists as a well-defined measure-valued process solving the regularized stochastic heat equation.
    Invoked to state the object whose large-time strong-disorder behavior is analyzed.
  • domain assumption The disorder distribution satisfies moment conditions allowing change-of-measure arguments to be applied uniformly.
    Required for the coarse-graining and extinction proofs to hold in the strong-disorder regime.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 2 internal anchors

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