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arxiv: 2605.02430 · v1 · submitted 2026-05-04 · 🧮 math.PR

Scaling limit of the range of tree-valued branching random walks in random environmen

Thomas Duquesne , Robin Khanfir This is my paper

Pith reviewed 2026-05-08 18:27 UTC · model grok-4.3

classification 🧮 math.PR
keywords branching random walkrandom treescaling limitBrownian cactusGromov-Hausdorff-Prokhorov convergencestable branching mechanismGalton-Watson treeoccupation measure
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The pith

Conditionally on the random tree, the scaled range of the branching random walk converges to the Brownian cactus with α-stable branching.

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

The paper shows that a critical biased branching random walk on a random supercritical Galton-Watson tree, indexed by a critical GW tree of size n, has its range converge after suitable scaling. Conditionally on the environment tree, the range equipped with rescaled graph distance and normalized occupation measure converges in the Gromov-Hausdorff-Prokhorov sense to the Brownian cactus with α-stable branching mechanism. This matters because it demonstrates that the scaling limit previously known for regular trees continues to hold when the underlying tree is itself random, under the condition that the offspring distribution belongs to the domain of attraction of an α-stable law for α in (1,2] with an extra moment assumption.

Core claim

Conditionally given the environment T, the measured metric space (R_n, s_n^{-1} d_gr, (1/n) m_occ^{(n)}) weakly converges in the Gromov-Hausdorff-Prokhorov sense to the Brownian cactus with α-stable branching mechanism.

What carries the argument

The measured metric space formed by the range R_n together with its graph distance and occupation measure, which is shown to converge after rescaling by s_n to the Brownian cactus.

If this is right

  • The same limit object appears whether the environment tree is regular or random.
  • The occupation measure of the range is normalized by 1/n in the limit.
  • The graph distance on the range must be scaled by a factor s_n that tends to infinity.
  • The convergence is weak convergence of measured metric spaces in the Gromov-Hausdorff-Prokhorov topology.
  • The result extends the earlier theorem for deterministic regular trees to the random-environment setting.

Where Pith is reading between the lines

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

  • The proof technique may adapt to other bias parameters or to branching random walks with different transition rules on the same class of random trees.
  • One can test whether removing the moment assumption on the offspring distribution still yields the same limit or produces a different scaling regime.

Load-bearing premise

The offspring distribution must satisfy a moment condition and lie in the domain of attraction of an α-stable law with α in (1,2].

What would settle it

A computation or simulation for large n showing that the rescaled range metric space fails to converge to the Brownian cactus when the offspring distribution satisfies the domain-of-attraction condition but violates the moment assumption.

read the original abstract

We study a branching random walk (BRW) taking its values in a random tree $\bT$ (seen as a family tree) with an infinite line of ancestors that is a variant of a supercritical Galton--Watson (GW) tree with offspring distribution $\nu$. The transition probabilities of the BRW are those of a critical biased random walk on $\bT$: namely, the probability to move from $x$ to one of its $k_x$ children is $1/(\mathtt{m}_\nu+k_x)$ and the probability to move from $x$ to the direct parent of $x$ is $\mathtt{m}_\nu/(\mathtt{m}_\nu+k_x)$. Here $\ttm_\nu$ stands for the mean of $\nu$. The BRW is indexed by a critical GW tree conditioned to have $n$ {vertices} and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha \ino (1, 2]$. We denote by $\cR_n$ the range of the BRW, i.e., ~the set of all sites in $\bT$ visited by the BRW. Under a moment assumption for $\nu$, we prove that if we view $\cR_n$ as a random subtree of $\bT$ equipped with its graph distance $d_{\mathtt{gr}}$ and with its occupation measure $\ttm^{_{(n)}}_{{\mathtt{occ}}}$ then there exists a scaling sequence $s_n \! \to \! \infty$ such that conditionally given the environment $\bT$, the measured metric space $(\cR_n, s_n^{-1}d_{\mathtt{gr}} , \frac{_1}{^n}\ttm^{_{(n)}}_{{\mathtt{occ}}} )$ weakly converges in the Gromov--Hausdorff--Prokhorov sense to a random measured compact real tree introduced by Curien, Le Gall \& Miermont in \cite{CuLGMi13} called the Brownian cactus with $\alpha$-stable branching mechanism. This work extends in random environment the result from D., K., Lin \& Torri \cite{DuKhLiTo22} which deals with the case where $\bT$ is a regular tree.

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

0 major / 3 minor

Summary. The manuscript proves a conditional scaling limit for the range of a branching random walk on a random supercritical Galton-Watson tree environment with infinite spine. Under a moment assumption on the offspring distribution ν (in the domain of attraction of an α-stable law, α ∈ (1,2]), the rescaled measured metric space (R_n, s_n^{-1} d_gr, (1/n) m_occ^{(n)}) converges weakly in the Gromov-Hausdorff-Prokhorov topology, conditionally on the environment, to the Brownian cactus with α-stable branching mechanism introduced by Curien-Le Gall-Miermont. The result extends the regular-tree case of DuKhLiTo22 by controlling the effect of random degrees via bias and the infinite spine.

Significance. If the technical arguments hold, the paper makes a solid contribution by extending scaling-limit results for branching random walks from deterministic regular trees to random environments. The conditional GHP convergence framework is appropriate and aligns with existing work on random trees and the Brownian cactus; the use of the infinite spine to anchor the environment is a natural and effective device. The result is falsifiable via simulation of the finite-n objects and provides a concrete link between discrete random media and continuum limits.

minor comments (3)
  1. The title contains a typographical error ('environmen' instead of 'environment').
  2. Notation for the mean m_ν and the occupation measure m_occ^{(n)} is introduced in the abstract but would benefit from an explicit reminder in the first paragraph of the introduction for readers who skip the abstract.
  3. The scaling sequence s_n is stated to exist and tend to infinity, but a brief indication of its growth rate (e.g., in terms of the stable index α) in the statement of the main theorem would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, for recognizing its contribution in extending the scaling limit result of DuKhLiTo22 from regular trees to random supercritical Galton-Watson environments with infinite spine, and for recommending minor revision. The conditional Gromov-Hausdorff-Prokhorov convergence to the Brownian cactus with α-stable branching mechanism is correctly captured in the referee's description.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is a conditional GHP convergence for the rescaled range of the BRW on a random tree environment, obtained by extending the regular-tree case via new controls on random degrees, bias, and the infinite spine. Tightness and limit identification rest on the stated moment assumption and domain-of-attraction condition for ν, which are independent of the prior work. The self-citation to DuKhLiTo22 is used only for the regular-tree baseline and does not bear the load of the random-environment argument; no step reduces by definition, fitted input, or self-citation chain to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard background from the theory of random trees, stable laws, and Gromov-Hausdorff-Prokhorov convergence; no new free parameters or invented entities are introduced beyond the scaling sequence s_n whose existence is asserted.

axioms (2)
  • standard math Gromov-Hausdorff-Prokhorov convergence is a valid topology for measured metric spaces
    Invoked for the weak-convergence statement in the abstract.
  • standard math The Brownian cactus with α-stable branching mechanism exists as a random measured real tree
    Cited from CuLGMi13 and used as the limit object.

pith-pipeline@v0.9.0 · 5736 in / 1371 out tokens · 37080 ms · 2026-05-08T18:27:58.473282+00:00 · methodology

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    A note on the Gromov-Hausdorff- Prokhorov distance between (locally) compact metric measure spaces.Electron

    ABRAHAM, R., DELMAS, J.-F.,ANDHOSCHEIT, P. A note on the Gromov-Hausdorff- Prokhorov distance between (locally) compact metric measure spaces.Electron. J. Probab. 18 (2013), no. 14, 21

    work page 2013

  2. [2]

    Scaling limit of the recurrent biased random walk on a Galton-Watson tree.Probab

    AÏDÉKON, E.,AND DERAPHÉLIS, L. Scaling limit of the recurrent biased random walk on a Galton-Watson tree.Probab. Theory Related Fields 169, 3-4 (2017), 643–666

    work page 2017

  3. [3]

    The continuum random tree

    ALDOUS, D. The continuum random tree. III.Ann. Probab. 21, 1 (1993), 248–289

    work page 1993

  4. [4]

    B.,ANDNEY, P

    ATHREYA, K. B.,ANDNEY, P. E.Branching processes. Springer-Verlag, New York- Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196

    work page 1972

  5. [5]

    On the trace of branching random walks.Groups Geom

    BENJAMINI, I.,ANDMÜLLER, S. On the trace of branching random walks.Groups Geom. Dyn. 6, 2 (2012), 231–247

    work page 2012

  6. [6]

    Cambridge Univ

    BERTOIN, J.Lévy Processes. Cambridge Univ. Press, 1996

    work page 1996

  7. [7]

    BIGGINS, J. D. Martingale convergence in the branching random walk.J. Appl. Probability 14, 1 (1977), 25–37

    work page 1977

  8. [8]

    Wiley Series in Probability and Mathematical Statistics

    BILLINGSLEY, P.Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, 1968

    work page 1968

  9. [9]

    H., GOLDIES, C

    BINGHAM, N. H., GOLDIES, C. M.,ANDTEUGELS, J. L.Regular Variation. Encyclopaedia of mathematics and its applications; 27. Cambridge University Press, 1987

    work page 1987

  10. [10]

    Brownian bridge on hyperbolic spaces and on homogeneous trees.Probab

    BOUGEROL, P.,ANDJEULIN, T. Brownian bridge on hyperbolic spaces and on homogeneous trees.Probab. Theory Related Fields 115, 1 (1999), 95–120

    work page 1999

  11. [11]

    BRAMSON, M. D. Maximal displacement of branching Brownian motion.Comm. Pure Appl. Math. 31, 5 (1978), 531–581

    work page 1978

  12. [12]

    BRAMSON, M. D. Minimal displacement of branching random walk.Z. Wahrsch. Verw. Gebiete 45, 2 (1978), 89–108

    work page 1978

  13. [13]

    BURKHOLDER, D. L. Martingale Transforms.The Annals of Mathematical Statistics 37, 6 (1966), 1494 – 1504

    work page 1966

  14. [14]

    Long Brownian bridges in hyperbolic spaces converge to Brownian trees.Electron

    CHEN, X.,ANDMIERMONT, G. Long Brownian bridges in hyperbolic spaces converge to Brownian trees.Electron. J. Probab. 22(2017), Paper No. 58, 15

    work page 2017

  15. [15]

    The Brownian cactus I

    CURIEN, N., LEGALL, J.-F.,ANDMIERMONT, G. The Brownian cactus I. Scaling limits of discrete cactuses.Ann. Inst. Henri Poincaré Probab. Stat. 49, 2 (2013), 340–373

    work page 2013

  16. [16]

    Central limit theorem for biased random walk on multi-type Galton-Watson trees.Electron

    DEMBO, A.,ANDSUN, N. Central limit theorem for biased random walk on multi-type Galton-Watson trees.Electron. J. Probab. 17(2012), no. 75, 40

    work page 2012

  17. [17]

    DRESS, A., MOULTON, V.,ANDTERHALLE, W.T-theory: an overview.European J. Combin. 17, 2-3 (1996), 161–175. Discrete metric spaces (Bielefeld, 1994)

    work page 1996

  18. [18]

    A limit theorem for the contour process of conditioned Galton-Watson trees

    DUQUESNE, T. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31, 2 (2003), 996–1027. 58

    work page 2003

  19. [19]

    Continuum tree limit for the range of random walks on regular trees.Ann

    DUQUESNE, T. Continuum tree limit for the range of random walks on regular trees.Ann. Probab. 33, 6 (2005), 2212–2254

    work page 2005

  20. [20]

    Scaling limits of tree-valued branch- ing random walks.Electronic Journal of Probability 27, none (2022), 1 – 54

    DUQUESNE, T., KHANFIR, R., LIN, S.,ANDTORRI, N. Scaling limits of tree-valued branch- ing random walks.Electronic Journal of Probability 27, none (2022), 1 – 54

    work page 2022

  21. [21]

    Random trees, Lévy processes and spatial branching processes.Astérisque, 281 (2002), vi+147

    DUQUESNE, T.,ANDLEGALL, J.-F. Random trees, Lévy processes and spatial branching processes.Astérisque, 281 (2002), vi+147

    work page 2002

  22. [22]

    Probabilistic and fractal aspects of Lévy trees.Probab

    DUQUESNE, T.,ANDLEGALL, J.-F. Probabilistic and fractal aspects of Lévy trees.Probab. Theory Related Fields 131, 4 (2005), 553–603

    work page 2005

  23. [23]

    Cambridge University Press, Cambridge, 2019

    DURRETT, R.Probability: Theory and Examples, 5th ed ed. Cambridge University Press, Cambridge, 2019

    work page 2019

  24. [24]

    Martingale convergence to mixtures of infinitely divisible laws.The Annals of Probability 3(06 1975)

    EAGLESON, G. Martingale convergence to mixtures of infinitely divisible laws.The Annals of Probability 3(06 1975)

    work page 1975

  25. [25]

    N.Probability and real trees, vol

    EVANS, S. N.Probability and real trees, vol. 1920 ofLecture Notes in Mathematics. Springer, Berlin, 2008. Lectures from the 35th Summer School on Probability Theory held in Saint- Flour, July 6–23, 2005

    work page 1920

  26. [26]

    GOUËZEL, S.,ANDLALLEY, S. P. Random walks on co-compact Fuchsian groups.Ann. Sci. Éc. Norm. Supér. (4) 46, 1 (2013), 129–173 (2013)

    work page 2013

  27. [27]

    Communication and Behavior

    HALL, P.,ANDHEYDE, C.Martingale Limit Theory and Its Application. Communication and Behavior. Academic Press, 1980

    work page 1980

  28. [28]

    HUETER, I.,ANDLALLEY, S. P. Anisotropic branching random walks on homogeneous trees.Probab. Theory Related Fields 116, 1 (2000), 57–88

    work page 2000

  29. [29]

    N.Limit theorems for stochastic processes, second ed., vol

    JACOD, J.,ANDSHIRYAEV, A. N.Limit theorems for stochastic processes, second ed., vol. 288 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2003

    work page 2003

  30. [30]

    Convergence of discrete snakes.J

    JANSON, S.,ANDMARCKERT, J.-F. Convergence of discrete snakes.J. Theoret. Probab. 18, 3 (2005), 615–647

    work page 2005

  31. [31]

    A simple proof of Duquesne’s theorem on contour processes of condi- tioned Galton-Watson trees.Séminaire de Probabilités, Lecture Notes in Mathematics, XLV (2013), 3126–3172

    KORTCHEMSKI, I. A simple proof of Duquesne’s theorem on contour processes of condi- tioned Galton-Watson trees.Séminaire de Probabilités, Lecture Notes in Mathematics, XLV (2013), 3126–3172

    work page 2013

  32. [32]

    De Gruyter, Berlin, New York, 1985

    KRENGEL, U.Ergodic Theorems. De Gruyter, Berlin, New York, 1985

    work page 1985

  33. [33]

    LALLEY, S. P. The weak/strong survival transition on trees and nonamenable graphs. In International Congress of Mathematicians. Vol. III. Eur. Math. Soc., Zürich, 2006, pp. 637– 647

    work page 2006

  34. [34]

    P.,ANDSELLKE, T

    LALLEY, S. P.,ANDSELLKE, T. Hyperbolic branching Brownian motion.Probab. Theory Related Fields 108, 2 (1997), 171–192

    work page 1997

  35. [35]

    A class of path-valued Markov processes and its applications to superpro- cesses.Probab

    LEGALL, J.-F. A class of path-valued Markov processes and its applications to superpro- cesses.Probab. Theory Related Fields 95, 1 (1993), 25–46

    work page 1993

  36. [36]

    Itô’s excursion theory and random trees.Stoch

    LEGALL, J.-F. Itô’s excursion theory and random trees.Stoch. Process. Appl., 120 (2010), 721–749. 59

    work page 2010

  37. [37]

    The Brownian cactus II: upcrossings and local times of super-Brownian motion.Probab

    LEGALL, J.-F. The Brownian cactus II: upcrossings and local times of super-Brownian motion.Probab. Theory Related Fields 162, 1-2 (2015), 199–231

    work page 2015

  38. [38]

    Branching processes in Lévy processes: the exploration process.Ann

    LEGALL, J.-F.,ANDLEJAN, Y. Branching processes in Lévy processes: the exploration process.Ann. Probab. 26, 1 (1998), 213–252

    work page 1998

  39. [39]

    Scaling limits of random planar maps with large faces

    LEGALL, J.-F.,ANDMIERMONT, G. Scaling limits of random planar maps with large faces. Ann. Probab., 39 (2011), 1–69

    work page 2011

  40. [40]

    LIGGETT, T. M. Branching random walks and contact processes on homogeneous trees. Probab. Theory Related Fields 106, 4 (1996), 495–519

    work page 1996

  41. [41]

    Random walks and percolation on trees.Ann

    LYONS, R. Random walks and percolation on trees.Ann. Probab. 18(1990), 931–958

    work page 1990

  42. [42]

    42 ofCambridge Series in Statistical and Probabilistic Mathematics

    LYONS, R.,ANDPERES, Y.Probability on trees and networks, vol. 42 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York, 2016

    work page 2016

  43. [43]

    States Spaces of the Snake and Its Tour—Convergence of the Discrete Snake.Journal of Theoretical Probability 16, 4 (2003), 1015 – 1046

    MARCKERT, J.-F.,ANDMOKKADEM, A. States Spaces of the Snake and Its Tour—Convergence of the Discrete Snake.Journal of Theoretical Probability 16, 4 (2003), 1015 – 1046

    work page 2003

  44. [44]

    Scaling limits of discrete snakes with stable branching.Ann

    MARZOUK, C. Scaling limits of discrete snakes with stable branching.Ann. Inst. Henri Poincaré Probab. Stat. 56, 1 (2020), 502–523

    work page 2020

  45. [45]

    A central limit theorem for biased random walks on Galton- Watson trees.Probab

    PERES, Y.,ANDZEITOUNI, O. A central limit theorem for biased random walks on Galton- Watson trees.Probab. Theory Related Fields 140, 3-4 (2008), 595–629

    work page 2008

  46. [46]

    REVUZ, D.,ANDYOR, M.Continuous Martingales and Brownian Motion, third edition ed., vol. 293. Springer, 1999

    work page 1999

  47. [47]

    2151 ofLecture Notes in Mathematics

    SHI, Z.Branching random walks, vol. 2151 ofLecture Notes in Mathematics. Springer, Cham, 2015. Lecture notes from the 42nd Probability Summer School held in Saint Flour, 2012, École d’Été de Probabilités de Saint-Flour

    work page 2015

  48. [48]

    Graduate texts in Mathematics

    SPITZER, F.Principles of Random Walks, 2d ed. Graduate texts in Mathematics. Springer, 1976

    work page 1976

  49. [49]

    ProQuest LLC, Ann Arbor, MI, 2016

    STEWART, A.On the Range of the Random Walk Bridge on the Regular Tree. ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of Toronto (Canada)

    work page 2016

  50. [50]

    Some useful functions for functional limit theorems.Math

    WHITT, W. Some useful functions for functional limit theorems.Math. Oper. Res. 5, 1 (1980), 67–85. 60

    work page 1980