Anomalous scaling limit of a Brownian particle in a log-correlated potential

Scott Armstrong, Ahmed Bou-Rabee, Tuomo Kuusi

Oct 8, 2026arXiv:2610.12335v1

math.PRmath-phmath.AP

Scorecard· 16/16

8.5/10 impact

Resolves a long-standing physics prediction and an open problem with a unique non-Gaussian singular scaling limit, via a deep and technically formidable synthesis of homogenization, RG, and LQG-type uniqueness.

Abstract

We study a Brownian particle in $\mathbb{R}^d$ , $d\geq2$ , with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.

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Scientific Impact Assessment: Anomalous scaling limit of a Brownian particle in a log-correlated potential

Core Contribution

The paper proves that a Brownian particle in ℝ^d (d≥2) drifting along the gradient of a weak-disorder log-correlated random potential has a unique scaling limit.

Limit process. It is a continuous, conservative strong Markov process, reversible with respect to a Gaussian multiplicative chaos (GMC) measure.

Singularity. The limit's path law is mutually singular with every mixture of Brownian laws. Its marginals are non-Gaussian, and its reversible measure is singular with respect to Lebesgue measure.

Anomalous scaling. The effective diffusivity decays as a power of scale. In d=2 the exponent is exact: a_m = e^{-(m+1)τ²}, from planar duality. In general d it matches the Bouchaud–Comtet–Georges–Le Doussal prediction to leading order in the disorder strength.

Open problem. The result addresses a long-standing physics prediction (Bouchaud et al. 1987) and answers Biskup's Problem 16.13 in the continuum setting.

Technically, the paper combines two sets of tools.

Renormalization-group induction. It uses Armstrong–Kuusi coarse-grained matrices, with errors measured relative to a decaying effective diffusivity and with no uniform ellipticity.

Liouville-quantum-gravity-style uniqueness. It adapts the Gwynne–Miller and Ding–Gwynne uniqueness arguments, with energy minimizers and harmonic replacement in place of geodesics, to show the limiting Dirichlet form is unique. The Devlin-style comparison of two cutoffs then upgrades convergence to almost sure.

The lower-bound proof of the effective diffusivity via null-Lagrangian duality in all dimensions (Prop. 5.7) is a neat additional piece.

Methodological Rigor

The proof architecture is coherent and covers a very long chain of steps.

It establishes uniform-in-cutoff coarse-graining estimates and a large-scale Hölder theorem with a Liouville theorem.

It proves compactness of the Dirichlet forms by Efron–Stein resampling.

It shows uniqueness via optimal comparison constants and a strict-improvement argument, then fixes the proportionality constant by homogenization.

It then handles tightness, resolvent convergence, and almost-sure convergence.

The honest accounting of limitations is a rigor strength. The renormalization error is O(δ) rather than vanishing, which is why uniqueness needs the LQG-type argument. The disorder threshold depends on the moment orders, and the general-d exponent is known only up to higher-order error.

The paper reports that Theorems A–C were formalized and machine-checked in Lean 4, though the repository is private. If verified, this is notable for a ~200-page argument. The paper also discloses extensive AI use for checking and writing. That is unusual, and it raises both a verification-confidence plus and a provenance question.

Potential Impact

The impact is mainly theoretical, in probability and PDE.

Subfields served. It opens rigorous study of diffusions in log-correlated environments, links stochastic homogenization to GMC and LQG-type techniques, and feeds the lattice analogue (random walks driven by the discrete Gaussian free field).

Reusable methods. The multiscale renormalization with relative errors, the harmonic-replacement uniqueness argument, and the cutoff-uniform regularity estimates are all reusable.

Applications. There are none direct. The model is an idealization from statistical physics.

Timeliness & Relevance

The result is well timed. LQG metrics, Liouville Brownian motion, and GMC are active areas. This is the first rigorous scaling limit for a genuinely non-Liouville diffusion in a log-correlated field. The Armstrong–Bou-Rabee–Kuusi program on critical random drifts and renormalization is also active.

Strengths

It settles a physics-level prediction rigorously, with an exact exponent in d=2.

It makes a clean conceptual statement: a unique, non-Gaussian, singular limit.

It brings together homogenization, renormalization, and LQG machinery.

It covers all d≥2, with non-Gaussian fields allowed.

The lower bound a_m ≥ e^{-(m+1)τ²} holds for all δ.

Limitations

The result is weak-disorder only. The strong-disorder and freezing regime (Castillo–Le Doussal) is untouched.

The general-d exponent is not exact.

The text is extremely long and technical, which limits accessibility and the ability to check it.

Relevant scales and moment thresholds make quantitative δ₀ non-explicit.

No numerical or physical validation is offered.

The paper is a major technical achievement with high expected impact within probability and mathematical physics, but it is narrow in direct applicability and extremely heavy. Clarity suffers somewhat from length, though each section has a good roadmap.

Rating:8.5/ 10

Significance 8.5Rigor 8.5Novelty 8.5Clarity 6

Generated Oct 9, 2026

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Scorecard· 16/16

8.5/10 impact