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arxiv: 2507.20324 · v2 · submitted 2025-07-27 · 🧮 math.PR

Non-existence of several random fractals in Brownian motion and Brownian loop soup

Yifan Gao , Xinyi Li , Runsheng Liu , Wei Qian This is my paper

Pith reviewed 2026-05-19 02:14 UTC · model grok-4.3

classification 🧮 math.PR MSC 60J6528A80
keywords Brownian motionloop soupHausdorff dimensionpioneer pointscut pointsintersection exponentsdisconnection exponents
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The pith

Exponents prove that pioneer triple points in planar Brownian motion do not exist.

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

This paper develops a unified approach to prove the non-existence of three kinds of random fractals in Brownian motion and loop soup by showing they have Hausdorff dimension zero. The fractals in question are the pioneer triple points of planar Brownian motion, the pioneer double cut points of planar and three-dimensional Brownian motions, and the double points on the boundaries of clusters in the planar Brownian loop soup at critical intensity. These results rely on intersection and disconnection exponents that give dimension exactly zero, implying the sets are empty almost surely. This matters because it answers open questions about whether these exceptional points can occur in the random structures.

Core claim

The non-existence of the pioneer triple points of the planar Brownian motion, the pioneer double cut points of the planar and three-dimensional Brownian motions, and the double points on the boundaries of the clusters of the planar Brownian loop soup at the critical intensity is established via a unified approach using intersection or disconnection exponents that yield Hausdorff dimension exactly zero.

What carries the argument

Intersection or disconnection exponents yielding Hausdorff dimension exactly zero for the associated point sets.

Load-bearing premise

The intersection and disconnection exponents from the literature are accurate and imply Hausdorff dimension exactly zero for these point sets without additional conditions.

What would settle it

A calculation or observation showing that any of these point sets has positive Hausdorff dimension in Brownian motion or the loop soup would disprove the non-existence.

Figures

Figures reproduced from arXiv: 2507.20324 by Runsheng Liu, Wei Qian, Xinyi Li, Yifan Gao.

Figure 1
Figure 1. Figure 1: U is the union of the curves. The opening is in green, the cut set on the intermediate scale is in blue and the “dead end” is in red. For later use, we also need to define the cut set with respect to a connected set in a general annulus. Let D1 ⊆ D2 ⊆ D3 be three discs (possibly with different centres). Let K0 be a 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The “links” and the “crossings” in the aforementioned decomposition, where the [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of the “tubes” (in the intermediate region), where the outer circle is D−m+3(z), and the two inner circles are D−m−2(S) and D−m−2(T). Finally, we give an upper bound for the third moment of the δ-good boxes. Lemma 3.9. For all three cases, there exists c > 0, such that for all n ≥ N(δ) and any n-boxes S, T, U ⊂ A(0, ι, 1 − ι) such that S, T, U ∈ Sn, we have P(S, T, U ∈ Sn) ≤ cD−d 1 D −d 2 2 −3dn… view at source ↗
Figure 4
Figure 4. Figure 4: The decomposition of the PTP case, where the “bridges” are dotted. The following lemma describes the law of Xi and Wi (see e.g. [12, Lemma 5.4]). Lemma 4.2. Given S ∈ Sn, on the event V PTP δ (S), the followings hold. (1) The joint law of W1 , . . . , W5 is uniformly equivalent to the joint law of five independent Brownian excursions in A(S, 2 −n , δ/2). (2) Conditioned on Xi , i = 0, 1, . . . , 5, W1 , . … view at source ↗
Figure 5
Figure 5. Figure 5: An example of the decomposition on the event E1,j . We omit W3, W4, W5 in the figure. Let t ′ i := τi(CjL), t′′ i := σi(C(j+1)L), i = 1, 2, . . . , 5. (4.10) Then E1,j ∩ E1 implies the following three events: (F 3 ) The union of Wi [0, t′ i ], i = 1, . . . , 5, does not disconnect D0 from infinity, (F 4 ) The union of Wi [t ′′ i , τi ], i = 1, . . . , 5, does not disconnect C(j+1)L from infinity. (F 5 ) Th… view at source ↗
Figure 6
Figure 6. Figure 6: An illustration for the proof of “only if” direction. The sets Oi+2, O′ are in blue. The three discs in this figure are C−n+(i+1)L(S), C−n+(i+2)L(S) and C−K(S). The paths Wi ’s (after hitting Pj ’s) are dashed. The path(s) passing through P1 or P4 are trapped in D (which is in green). Finally, note that everything mentioned above is fully determined by U(i+3)L, the result hence follows. 4.3 Proof of Propos… view at source ↗
Figure 7
Figure 7. Figure 7: An illustration for the proof of “only if” direction. The dashed curves are (pieces of) W m. Oi+2, O′ are in blue. The four circles in this figure are C−n+(i+1)L(S), C−n+(i+2)L(S), C−n+(i+3)L(S) and C−K(S). Then, after hitting P1 or the loop cluster passing through P1, Wj1 is trapped in the bounded domain D. Finally, note that dist(S, T) ≤ 2 −n+(i+1)L < 2 − √ n , hence {S ∈ Sn(δ + 2− √ n )} implies that {T… view at source ↗
read the original abstract

We develop a unified approach to establish the non-existence of three types of random fractals: (1) the pioneer triple points of the planar Brownian motion, answering an open question in [7], (2) the pioneer double cut points of the planar and three-dimensional Brownian motions, and (3) the double points on the boundaries of the clusters of the planar Brownian loop soup at the critical intensity, answering an open question in [39]. These fractals have the common feature that they are associated with an intersection or disconnection exponent which yields a Hausdorff dimension ``exactly zero''.

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 develops a unified approach to prove the almost-sure non-existence of three families of random fractals associated with Brownian motion and the Brownian loop soup: (1) pioneer triple points of planar Brownian motion, (2) pioneer double cut points of planar and three-dimensional Brownian motion, and (3) double points on the boundaries of clusters in the critical planar Brownian loop soup. Each non-existence statement is obtained by associating the set with an intersection or disconnection exponent from the literature whose corresponding Hausdorff-dimension formula evaluates to exactly zero.

Significance. Resolving the open questions cited from [7] and [39] would be a useful contribution to the study of exceptional points of Brownian paths and loop soups. A unified exponent-based method that simultaneously treats several dimension-zero cases is potentially reusable and therefore of interest to the field, provided the critical case is handled rigorously.

major comments (1)
  1. [§1 and main theorems] §1 and the proofs of the main theorems: the manuscript concludes non-existence from the fact that the relevant intersection/disconnection exponents produce Hausdorff dimension exactly zero. The standard implication dim_H < 0 ⇒ empty a.s. does not automatically extend to the boundary case dim_H = 0; logarithmic corrections or capacity arguments may be required. An explicit lemma or reference establishing emptiness in this critical regime must be supplied (or the existing argument strengthened) because this step is load-bearing for all three claims.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should spell out the precise dimension formula (e.g., 2 − ξ or its analogue) used for each of the three sets so that the reader can immediately verify that the exponent indeed yields zero.
  2. [Throughout] Notation for the various exponents (intersection, disconnection, pioneer, etc.) should be collected in a short table or glossary to avoid repeated re-definition across sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to rigorously justify the critical case of Hausdorff dimension exactly zero. We address the major comment below and will incorporate the suggested strengthening into the revised version.

read point-by-point responses

  1. Referee: [§1 and main theorems] §1 and the proofs of the main theorems: the manuscript concludes non-existence from the fact that the relevant intersection/disconnection exponents produce Hausdorff dimension exactly zero. The standard implication dim_H < 0 ⇒ empty a.s. does not automatically extend to the boundary case dim_H = 0; logarithmic corrections or capacity arguments may be required. An explicit lemma or reference establishing emptiness in this critical regime must be supplied (or the existing argument strengthened) because this step is load-bearing for all three claims.

    Authors: We agree that the implication for the boundary case dim_H = 0 does not follow automatically from the strict inequality and requires explicit justification. In the revised manuscript we will insert a new short lemma (placed in the preliminaries section) that handles this critical regime uniformly for the three settings. The lemma combines a first-moment argument on the number of approximating points with a capacity estimate derived from the known intersection and disconnection exponents; when the exponent yields dimension zero the capacity vanishes, implying the set is empty almost surely. This argument is standard in the literature on Brownian exponents (see e.g. the capacity methods in Lawler’s book and the recent works on critical SLE dimensions) and applies directly to the pioneer triple points, pioneer double cut points, and loop-soup boundary double points. We will also add a brief remark in §1 explaining why logarithmic corrections do not appear in these particular exponent calculations. The change affects only the justification step and leaves the main theorems and their proofs otherwise unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on intersection and disconnection exponents drawn from prior external literature to obtain Hausdorff dimension exactly zero, then concludes non-existence of the indicated point sets. No step in the provided abstract or described unified approach reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain internal to the manuscript. The central claims address open questions by applying established exponent results, rendering the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard properties of Brownian motion, loop soups, and known intersection/disconnection exponents from prior work; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Known values of intersection and disconnection exponents for Brownian motion and loop soup yield Hausdorff dimension exactly zero for the target sets.
    Invoked to conclude non-existence from dimension calculation.

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    O. Adelman. Brownian motion never increases: A new proof to a result of Dvoretzky, Erd˝ os and Kakutani. Israel J. Math. , 1985, 50(3): 189–192

    work page 1985

  2. [2]

    R. F. Bass, K. Burdzy. Cutting Brownian paths. American Mathematical Soc., 1999

    work page 1999

  3. [3]

    S. Berman. Nonincrease almost everywhere of certain measurable functions with applica- tions to stochastic processes. Proc. Amer. Math. Soc., 1983, 88(1):141–144

    work page 1983

  4. [4]

    J. Bertoin. Increase of a L´ evy process with no positive jumps. Stoch. Stoch. Rep. , 1991, 37(4):247–251

    work page 1991

  5. [5]

    K. Burdzy. On nonincrease of Brownian motion. Ann. Probab., 1990, 18(3): 978–980

    work page 1990

  6. [6]

    Burdzy, G

    K. Burdzy, G. F. Lawler. Non-intersection exponents for Brownian paths: Part I. Existence and an invariance principle. Probab. Theory Related Fields, 1990, 84(3): 393–410

    work page 1990

  7. [7]

    Burdzy, W

    K. Burdzy, W. Werner. No triple point of planar Brownian motion is accessible. Ann. Probab., 1996, 24(1): 125–147

    work page 1996

  8. [8]

    G. Cai, Y. Gao. Uniqueness of generalized conformal restriction measures and Malliavin- Kontsevich-Suhov measures for c ∈ (0, 1]. Preprint, available at arXiv:2502.05890

    work page arXiv

  9. [9]

    Duplantier, K

    B. Duplantier, K. H. Kwon. Conformal invariance and intersections of random walks. Phys. Rev. Lett., 1988, 61(22): 2514

    work page 1988

  10. [10]

    Dvoretzky, P

    A. Dvoretzky, P. Erd˝ os, S. Kakutani, S. J. Taylor. Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 1957(53): 856–862

    work page 1957

  11. [11]

    Dvoretzky, P

    A. Dvoretzky, P. Erd˝ os, S. Kakutani. Nonincrease everywhere of the Brownian motion process. Proc. 4th Berkeley Sympos. Math. Statist. and Prob. , 1961, 2: 103–116

    work page 1961

  12. [12]

    Y. Gao, X. Li, W. Qian. Multiple points on the boundaries of Brownian loop-soup clusters. To appear in Ann. Probab., also available at arXiv:2205.11468

    work page arXiv

  13. [13]

    Y. Gao, P. Nolin, W. Qian. Up-to-constants estimates on four-arm events for simple con- formal loop ensemble. Preprint, available at arXiv:2504.06202

    work page arXiv

  14. [14]

    Holden, G

    N. Holden, G. F. Lawler, X. Li, X. Sun. Minkowski content of Brownian cut points. Ann. Inst. Henri Poincar´ e, Probab. Stat., 2022, 58(1): 455–488

    work page 2022

  15. [15]

    A. Jego, T. Lupu, W. Qian. Crossing exponent in the Brownian loop soup. To appear in Ann. Probab., also available at arXiv:2303.03782

    work page arXiv

  16. [16]

    Kiefer, P

    R. Kiefer, P. M¨ orters. The Hausdorff dimension of the double points on the Brownian frontier. J. Theor. Probab., 2010, 23: 605–623

    work page 2010

  17. [17]

    F. B. Knight. Essentials of Brownian motion and diffusion. American Mathematical Soc., 1981

    work page 1981

  18. [18]

    G. F. Lawler. Intersections of random walks with random sets. Israel J. Math., 1989, 65(2): 113–132

    work page 1989

  19. [19]

    G. F. Lawler. Nonintersecting planar Brownian motions. Math. Phys. Electron. J. , 1995, 1(4): 1–35

    work page 1995

  20. [20]

    G. F. Lawler. Hausdorff dimension of cut points for Brownian Motion. Electron. Comm. Probab., 1996, 1: 1–20

    work page 1996

  21. [21]

    G. F. Lawler. The Dimension of the Frontier of Planar Brownian Motion. Electron. Comm. Probab., 1996, 1: 29–47. 45

    work page 1996

  22. [22]

    G. F. Lawler. Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J. , 1998, 4(5): 1–67

    work page 1998

  23. [23]

    G. F. Lawler. Conformally invariant processes in the plane , volume 114 of Mathematical Surveys and Monographs. American Mathematical Soc, Providence, RI, 2005

    work page 2005

  24. [24]

    G. F. Lawler. Intersections of Random Walks , Birkh¨ auser New York, 2013

    work page 2013

  25. [25]

    G. F. Lawler and V. Limic. Random Walk: A Modern Introduction , Cambridge University Press, 2010

    work page 2010

  26. [26]

    G. F. Lawler, B. Vermesi. Fast convergence to an invariant measure for non-intersecting 3- dimensional Brownian paths. ALEA, Lat. Am. J. Probab. Math. Stat. , 2012, 9(2): 717–738

    work page 2012

  27. [27]

    G. F. Lawler, O. Schramm, W. Werner. The Dimension of the Planar Brownian Frontier is 4/3. Math. Res. Lett., 2001, 8(4): 401–411

    work page 2001

  28. [28]

    G. F. Lawler, O. Schramm, W. Werner. Values of Brownian intersection exponents, I: Half- plane exponents. Acta Math., 2001, 187(2): 237–273

    work page 2001

  29. [29]

    G. F. Lawler, O. Schramm, W. Werner. Values of Brownian intersection exponents, II: Plane exponents. Acta Math., 2001, 187(2): 275–308

    work page 2001

  30. [30]

    G. F. Lawler, O. Schramm, W. Werner. Values of Brownian intersection exponents, III: Two-sided exponents. Ann. Inst. Henri Poincar´ e, Probab. Stat., 2002, 38(1): 109–123

    work page 2002

  31. [31]

    G. F. Lawler, O. Schramm, W. Werner. Analyticity of intersection exponents for planar Brownian motion. Acta Math., 2002, 189(2): 179–201

    work page 2002

  32. [32]

    G. F. Lawler, O. Schramm, W. Werner. Sharp estimates for Brownian non-intersection probabilities. In and Out of Equilibrium: Probability with a Physics Flavor. Birkh¨ auser Boston, 2002: 113–131

    work page 2002

  33. [33]

    G. F. Lawler, W. Werner. Intersection exponents for planar Brownian motion.Ann. Probab., 1999, 27: 1601–1642

    work page 1999

  34. [34]

    G. F. Lawler, W. Werner. The Brownian loop soup. Probab. Theory Related Fields , 2004, 128(4): 565–588

    work page 2004

  35. [35]

    Le Gall, T

    J.-F. Le Gall, T. Meyre. Points cˆ ones du mouvement brownien plan, le cas critique.Probab. Theory Related Fields, 1992, 93: 231–247

    work page 1992

  36. [36]

    where did the free planar bosons go?

    M. Lehmkuehler, W. Qian, W. Werner. Parity questions in critical planar Brownian loop- soups (or “where did the free planar bosons go?”). Preprint, available at arXiv:2403.07830

    work page arXiv

  37. [37]

    M¨ orters, Y

    P. M¨ orters, Y. Peres.Brownian motion. Cambridge University Press, 2010

    work page 2010

  38. [38]

    Y. Peres. Points of increase for random walks. Israel J. Math. , 1996, 95(1): 341–347

    work page 1996

  39. [39]

    W. Qian. Generalized disconnection exponents. Probab. Theory Related Fields, 2021, 179(1- 2): 117–164

    work page 2021

  40. [40]

    W. Qian, W. Werner. The law of a point process of Brownian excursions in a domain is determined by the law of its trace. Electron. J. Probab., 2018, 23(128): 1–23

    work page 2018

  41. [41]

    W. Qian, W. Werner. Decomposition of Brownian loop-soup clusters. J. Eur. Math. Soc. , 2019, 21(10): 3225–3253

    work page 2019

  42. [42]

    Rohde, O

    S. Rohde, O. Schramm. Basic properties of SLE. Ann. of Math. , 2005, 161(2): 883–924

    work page 2005

  43. [43]

    Sheffield, W

    S. Sheffield, W. Werner. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. , 2012, 176(3): 1827–1917

    work page 2012

  44. [44]

    B. Vir´ ag. Brownian beads.Probab. Theory Related Fields, 2003, 127(3): 367–387

    work page 2003

  45. [45]

    W. Werner. On the spatial Markov property of soups of unoriented and oriented loops. In S´ eminaire de Probabilit´ es XLVIII, volume 2168 of Lecture Notes in Math. , Springer, 2016: 481–503. 46

    work page 2016