Graph distance and percolation dimension of three-dimensional Brownian motion

Daisuke Shiraishi

Oct 23, 2026arXiv:2610.32137v1

math.PR

Scorecard· 15/16

8.0/10 impact

Resolves a decades-old percolation-dimension problem for 3D Brownian motion with a new graph-distance characterization and a metric-limit framework, though the exponent itself remains unidentified.

Abstract

We prove that the minimum Hausdorff dimension of a continuous curve joining $W_{0}$ to $W_{1}$ within the range $W ([0, 1])$ of a three-dimensional Brownian motion is almost surely a deterministic constant $γ_3\in(1,5/3]$ . Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension $γ_{3}$ and admits a parametrization that is $β$ -Hölder for every $0 < β < 1 / γ_{3}$ . To identify $γ_{3}$ , we partition Brownian time into $n$ equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent $θ$ and that $γ_{3} = 2 θ$ . We also establish $n^{pθ+o(1)}$ growth for their $p$ -th moments, for every $p > 0$ .

AI Impact Assessments

(1 model)

Paper type: Theoretical (probability / fractal geometry of Brownian motion). It is a long, technically heavy paper with no numerics.

Core Contribution

The paper settles the percolation-dimension problem for 3D Brownian motion, a question dating to Burdzy (1990) and Fargason (1998). It proves that the minimal Hausdorff dimension of a continuous curve from W_0 to W_1 inside the range is almost surely a deterministic constant γ_3 ∈ (1, 5/3]. Earlier work only gave the bounds 1 < p_3 < 2, with a strict lower bound from Lawler's cut points and a non-sharp upper bound from Fargason.

The main conceptual step is a discrete characterization of γ_3:

Partition Brownian time into n cells and join cells whose ranges intersect.

The endpoint distance and diameter of this graph then grow like n^{θ+o(1)} almost surely and in every moment, and γ_3 = 2θ.

A subadditivity argument (Fekete) gives existence of the exponent.

The upper bound on p_3 comes from a multiscale curve construction with a loop-removal lemma.

The lower bound comes from limiting Hölder functions that are compatible with the path, combined with Kaufman's dimension doubling.

Blumenthal's zero-one law upgrades positive probability to probability one.

Further results include:

Continuum many simple minimizing curves with distinct images, built from the absence of pioneer double cut points (Gao–Li–Liu–Qian).

Box dimension equal to γ_3, and a parametrization that is β-Hölder for every β < 1/γ_3.

A subsequential scaling limit of the normalized graph distances, giving a random pseudometric on Brownian time. It has a gluing formula and is a measurable function of the path, indeed of the self-intersection relation.

Separation of a deterministic time from all other times.

Ball-exit partitions with exponent γ_3.

The comparison γ_3 ≤ ℓ_3 ≤ 5/3, where ℓ_3 is the loop-erased random walk growth exponent.

Methodological Rigor

The arguments are detailed and appear careful.

The submultiplicative structure is handled with an exponential tail bound from the strong Markov property, and the log factors are managed correctly.

The Paley–Zygmund step for moments is standard and sound.

The lower-bound strategy is clever: a compatible Hölder function from the normalized graph distance, then dimension doubling.

Measurability is addressed explicitly, including analytic-set issues for p_3(W).

The conditional Efron–Stein argument for measurability of the limit (variance O(m^{1−2θ}), which relies on θ > 1/2) is neat.

The nonuniqueness construction is intricate but logically organized.

Weaknesses:

The exponent θ is never identified numerically; it is only bounded, so γ_3 ∈ (1, 5/3].

The limit metric is only subsequential; uniqueness is open.

Full spatial separation (D(s,t) = 0 iff W_s = W_t) is open, so the "every pair" statement holds only for fixed s or Lebesgue-almost every s.

The paper discloses use of LLMs for checking proofs. Human verification is claimed.

Its length (57 pages) makes full verification by referees demanding.

Potential Impact

This is a landmark-type result within probability and fractal geometry: it closes a 35-year-old question for d = 3. It also introduces a framework (the graph-distance exponent, the intrinsic metric, subsequential limits) that:

connects Brownian percolation dimension to loop-erased random walk, with γ_3 ≤ ℓ_3 (and the paper's own open questions ask whether γ_3 = 3/2);

opens a path to a "Brownian chemical-distance" metric analogous to Liouville or Brownian-map metrics;

suggests related questions for random walk traces in d = 2, 3, 4, 5 (cited companion preprints) and for planar Brownian motion.

It is unlikely to have effects outside mathematical probability and statistical physics, although the chemical-distance exponent is of interest to physicists studying random-walk clusters.

Timeliness & Relevance

The paper arrives amid active work on non-existence results for Brownian random fractals (Gao–Li–Liu–Qian), on 3D loop-erased random walk (Kozma, Shiraishi), and on random metrics. It uses recent results (pioneer double cut points) in an essential way, and it appears alongside a series of companion preprints on graph distance of random walk traces. The ideas are likely to be reused in these neighboring problems.

Strengths

Solves a named, long-open problem with an exact characterization, rather than just improved bounds.

Introduces a robust, reusable discrete-to-continuum scheme: Fekete-type exponent existence, compatible Hölder functions, and a zero-one law.

Includes extras that add value beyond the headline: nonuniqueness, a metric limit with gluing, and ball-exit partitions.

Clear logical structure, with an outline in the introduction and explicit open questions.

Limitations

γ_3 is not computed. Numerical bounds remain wide (1 < γ_3 ≤ 5/3), and the paper does not say whether γ_3 = ℓ_3 or 3/2.

The limiting metric is not shown to be unique or a true metric on the range.

The result is specific to d = 3 (the planar case is open).

Some parts are dense and notation-heavy.

Overall

A strong, technically deep probability paper with a high-profile resolution. Its impact is concentrated within probability, but within that community it is substantial.

Rating:8/ 10

Significance 7.5Rigor 8Novelty 8Clarity 6.5

Generated Oct 26, 2026

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Scorecard· 15/16

8.0/10 impact

Resolves a decades-old percolation-dimension problem for 3D Brownian motion with a new graph-distance characterization and a metric-limit framework, though the exponent itself remains unidentified.