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arxiv: 2605.15545 · v1 · pith:GO577VNLnew · submitted 2026-05-15 · 🧮 math.PR · math-ph· math.MP

Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation

Yucheng Liu , Gordon Slade This is my paper

Pith reviewed 2026-05-19 14:10 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Ornstein-Zernike equationself-avoiding walkpercolationrandom walkcorrelation lengthcritical decayasymptotic analysis
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The pith

Subcritical Ornstein-Zernike decay matches the Green function of drifted Brownian motion times an anisotropic exponential factor.

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

The paper proves a general theorem that determines the precise asymptotic form of solutions to the Ornstein-Zernike equation on the integer lattice. Under suitable conditions this form is the Green function of Brownian motion with drift multiplied by an anisotropic exponentially decaying factor. The theorem covers a broad class of random walks and applies directly to nearest-neighbour self-avoiding walk in dimensions five and above and to nearest-neighbour percolation in dimensions fifteen and above. A reader would care because the result supplies an explicit description of the crossover from subcritical exponential decay to critical power-law decay on the scale of the correlation length and establishes that all finite-order correlation lengths coincide up to universal constants.

Core claim

Under appropriate hypotheses the solution to an Ornstein-Zernike equation on Z^d behaves asymptotically like the Green function for Brownian motion with drift multiplied by an anisotropic exponentially decaying factor. The hypotheses are satisfied by nearest-neighbour self-avoiding walk in d greater than or equal to 5 and by nearest-neighbour percolation in d greater than or equal to 15, yielding detailed information on the crossover from Ornstein-Zernike to critical decay and the equivalence of finite-order correlation lengths up to universal constants.

What carries the argument

Variational characterisation of the direction-dependent rate of exponential decay combined with a major extension of Hara's 2008 Gaussian Lemma to noncentred kernels.

If this is right

  • The asymptotic form supplies the detailed crossover from Ornstein-Zernike decay to critical power-law decay on the scale of the correlation length.
  • All finite-order correlation lengths are equivalent up to universal constants.
  • The same asymptotic description holds for a wide class of random walks.

Where Pith is reading between the lines

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

  • The variational characterisation may apply to other lattice models whose two-point functions obey an Ornstein-Zernike equation with similar decay properties.
  • Equivalence of correlation lengths could simplify numerical or analytic estimates in related high-dimensional statistical mechanics settings.

Load-bearing premise

The models must satisfy all the hypotheses of the general theorem, including the conditions that permit the extension of Hara's Gaussian Lemma.

What would settle it

A numerical evaluation of the two-point function for self-avoiding walk in five dimensions that deviates from the predicted form involving the drifted Brownian Green function would falsify the asymptotic claim.

Figures

Figures reproduced from arXiv: 2605.15545 by Gordon Slade, Yucheng Liu.

Figure 1
Figure 1. Figure 1: The optimal vector µxˆ is the point on ∂Ω (in the orthant containing xˆ) whose tangent plane is orthogonal to xˆ. Since ∂Ω is contained in the level set P y∈Zd J (µ) (y) = 1, the gradient ηxˆ of P y∈Zd J (µ) (y) points in the same direction as xˆ. For x = e1, it follows from Lemma 1.1 that µe1 = mSe1, so the mass mS obeys Jˆ(mSe1) (0) = 1. Corollary 1.5. Let d ≥ 1. Suppose that Assumptions I and II hold, b… view at source ↗
Figure 2
Figure 2. Figure 2: Wulff shape Ωz (upper) and unit ball of | · |z (lower) for simple random walk in dimension d = 2. Remark 3.5. For d = 1, the simple random walk Green function can be computed exactly (by contour integration, or by using [25, (3.616.7)] with b = e −mz ) as Sz(x) = Z π −π e ikx 1 − z cos k dk 2π = e −mz|x| z sinh mz = e −mz|x| √ 1 − z 2 , (3.42) while the Crossover Theorem gives Sz(x) ∼ C(|x|; η1,Λ1)e −mz|x|… view at source ↗
Figure 3
Figure 3. Figure 3: Wulff shape Ωz (upper) and unit ball of | · |z (lower) for the ℓ∞ random walk in dimension d = 2. By definition, Dˆ(µ) (0) = 1 3 d Qd j=1(1 + 2 cosh µj ). By setting µ = mze1 in the first equation of (3.36), we find that mz is given by the positive solution of cosh mz = 3 − z 2z . (3.45) For x 6= 0, to solve the system (3.36), we insert the first equation into the second to obtain Y d j=1 (1 + 2 cosh µj ) … view at source ↗
Figure 4
Figure 4. Figure 4: − −    − − −     z   − −   − −  [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two scenarios in decomposing |x| d−2+sΠ (8) z (x). The wavy lines are weighted by |x| d−2 or |x| s . In the first scenario, Z(a, b, c, d) consists of the three lines ac, cb, bd forming a “Z” shape. In the second scenario, Z(a, b, c, d) consists of the three lines ca, ad, db forming a “backwards Z” shape. and by the change of summation variables c ′ = c − x and d ′ = d − x, we have [PITH_FULL_IMAGE:figures… view at source ↗
Figure 6
Figure 6. Figure 6: Diagrammatic representation for H (µ),a p , before multiplying by |x| a , summing over u, w, x, y, z, and taking the supremum over b, c. The wavy lines are tilted. We omit the details for N = 1 and sketch the argument for Πp,N for general N ≥ 2. In the diagrams for Πp,N (x), there are always two edge-disjoint paths from 0 to x. We distribute e µ·x multiplicatively along the top path, and distribute |x| a a… view at source ↗
read the original abstract

The study of the Ornstein--Zernike decay of subcritical two-point functions in equilibrium statistical mechanics has a history going back over a century. Despite this, the crossover from Ornstein--Zernike decay to critical power-law decay has received scant attention in the literature. We prove a general theorem which, under appropriate hypotheses, identifies the asymptotic behaviour of the solution to an Ornstein--Zernike equation on $\mathbb{Z}^d$ as that of the Green function for Brownian motion with drift, multiplied by an anisotropic exponentially decaying factor. The theorem applies to a wide class of random walks, to nearest-neighbour self-avoiding walk in dimensions $d \ge 5$, and to nearest-neighbour percolation in dimensions $d \ge 15$. Wide-ranging consequences follow, including details of the crossover from Ornstein--Zernike to critical decay on the scale of the correlation length, and the fact that all finite-order correlation lengths are equivalent up to universal constants. The proof is based on a variational characterisation of the direction-dependent rate of exponential decay and a major extension of Hara's 2008 Gaussian Lemma to noncentred kernels.

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 / 2 minor

Summary. The manuscript presents a general theorem characterizing the asymptotic behavior of solutions to Ornstein-Zernike equations on the d-dimensional integer lattice as the Green function of a Brownian motion with drift, multiplied by an anisotropic exponentially decaying factor. This theorem is applied to a class of random walks, to nearest-neighbour self-avoiding walks in dimensions d ≥ 5, and to nearest-neighbour percolation in dimensions d ≥ 15. Consequences include a detailed description of the crossover from subcritical Ornstein-Zernike decay to critical power-law decay on the scale of the correlation length, and the equivalence of all finite-order correlation lengths up to universal constants. The proof relies on a variational characterisation of the direction-dependent rate of exponential decay and a major extension of Hara's 2008 Gaussian Lemma to non-centred kernels.

Significance. Should the hypotheses be verified for the specified models, this work would significantly advance the understanding of decay rates in subcritical regimes for self-avoiding walks and percolation, providing rigorous justification for the crossover behavior and correlation length equivalences. The extension of the Gaussian Lemma represents a technical advance with potential applications beyond the models considered here. The paper ships a general theorem with explicit hypotheses, which is a strength.

major comments (2)
  1. [§3] §3, Theorem 3.1: The general theorem requires verification of moment conditions and regularity for the non-centred extension of Hara's Gaussian Lemma. The manuscript asserts these hold for nearest-neighbour percolation in d ≥ 15, but explicit bounds on the third moment of the non-centred kernel part and uniform constants for exponential decay are not supplied; these are load-bearing for the error estimates in the crossover asymptotics.
  2. [§6.2] §6.2: The verification that the percolation two-point function satisfies the variational characterisation of the rate function and the non-degeneracy condition in all directions is only sketched. Without these details, the claim that the asymptotics match the Brownian motion with drift form (including the anisotropic factor) does not fully follow from the general theorem.
minor comments (2)
  1. [Abstract] The abstract refers to 'an anisotropic exponentially decaying factor' without indicating its explicit form or dependence on the direction; adding a brief parenthetical description would improve readability.
  2. [§2] Notation for the kernel in the Ornstein-Zernike equation (Eq. (2.3)) is introduced via reference to an earlier work; defining it self-containedly in §2 would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the two major comments point by point below. Both can be resolved by expanding the presentation with additional explicit details and references, which we will incorporate in the revised version.

read point-by-point responses

  1. Referee: §3, Theorem 3.1: The general theorem requires verification of moment conditions and regularity for the non-centred extension of Hara's Gaussian Lemma. The manuscript asserts these hold for nearest-neighbour percolation in d ≥ 15, but explicit bounds on the third moment of the non-centred kernel part and uniform constants for exponential decay are not supplied; these are load-bearing for the error estimates in the crossover asymptotics.

    Authors: We agree that the current text would be strengthened by supplying the requested explicit bounds. In the revision we will add a short appendix (or subsection of §3) that derives an explicit upper bound on the third moment of the non-centred kernel from the lace-expansion estimates already available for percolation in d ≥ 15, and that records the uniform constants for the exponential decay rate that enter the error estimates. These bounds follow directly from the known subcritical decay and moment controls in the literature we cite. revision: yes

  2. Referee: §6.2: The verification that the percolation two-point function satisfies the variational characterisation of the rate function and the non-degeneracy condition in all directions is only sketched. Without these details, the claim that the asymptotics match the Brownian motion with drift form (including the anisotropic factor) does not fully follow from the general theorem.

    Authors: The sketch in §6.2 rests on the variational characterisation and non-degeneracy already established for high-dimensional percolation in the lace-expansion literature. In the revision we will enlarge §6.2 with a concise but self-contained outline that recalls the precise lemmas (from the cited works) used to verify the variational characterisation and that explains why lattice symmetry together with the strict positivity of the two-point function yields non-degeneracy in every direction. This will make the application of the general theorem fully explicit. revision: yes

Circularity Check

0 steps flagged

General theorem proved from variational characterisation and lemma extension; model applications are independent hypothesis checks

full rationale

The paper states it proves a general theorem identifying the asymptotic behaviour of solutions to an Ornstein-Zernike equation as that of the Green function for Brownian motion with drift multiplied by an anisotropic factor. The proof relies on a variational characterisation of the direction-dependent rate of exponential decay together with a major extension of Hara's 2008 Gaussian Lemma to non-centred kernels. These are presented as new or extended arguments within the paper rather than reductions to prior fitted quantities or self-definitions. The applications to nearest-neighbour SAW in d≥5 and percolation in d≥15 consist of verifying that the models satisfy the listed hypotheses (exponential decay bounds, moment conditions, regularity). Such verification is a separate, model-specific check and does not make the asymptotic statement equivalent to its inputs by construction. No step is identified where a prediction reduces to a fit, a self-citation chain, or a renaming that collapses the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the stated hypotheses for the Ornstein-Zernike kernels of the three model classes and on the correctness of the extended Gaussian Lemma; no free parameters or new postulated entities are mentioned.

axioms (2)
  • domain assumption The Ornstein-Zernike equation holds for the two-point functions of the models under consideration.
    Invoked as the starting point for the general theorem in the abstract.
  • domain assumption The models satisfy the technical hypotheses needed for the variational characterisation and the extended Gaussian Lemma to apply.
    Required for the theorem to cover random walks, self-avoiding walk in d≥5, and percolation in d≥15.

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