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arxiv: 2604.24389 · v1 · submitted 2026-04-27 · 🧮 math.PR · cond-mat.stat-mech· math-ph· math.MP

"True" self-avoiding walks on general trees

Tuan-Minh Nguyen This is my paper

Pith reviewed 2026-05-08 01:55 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechmath-phmath.MP
keywords true self-avoiding walkbranching-ruin numberrecurrencetransiencephase transitioninfinite treesHausdorff dimension
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The pith

True self-avoiding walks on trees switch from recurrent to transient exactly when the branching-ruin number exceeds 1/2.

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

The paper studies true self-avoiding random walks on infinite locally finite trees, where each edge's selection probability decays exponentially with the number of prior traversals. It establishes a sharp phase transition in the walk's long-term behavior governed by the tree's branching-ruin number. The walk returns to the root infinitely often almost surely below the critical value and escapes every finite set almost surely above it. This supplies an exact criterion for recurrence versus transience on arbitrary trees.

Core claim

The true self-avoiding walk with edge weights exp(-β n) after n traversals is almost surely recurrent when the branching-ruin number is less than 1/2 and almost surely transient when the branching-ruin number is greater than 1/2, with the critical value coinciding with the Hausdorff dimension of the tree boundary under a suitable metric.

What carries the argument

The branching-ruin number of the tree, which sets the threshold for the recurrence-transience transition by quantifying how the tree's growth competes with the exponential penalization of repeated edges.

Load-bearing premise

The branching-ruin number accurately locates the balance point between the tree's expansion and the exponential weight decay that decides whether the walk returns or escapes.

What would settle it

Compute or simulate the return probability of the walk on a tree with branching-ruin number known to be 0.6 and check whether it tends to zero (transient) or stays positive (recurrent).

read the original abstract

We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse with probability proportional to the current weight of that edge, where the weight of each edge after being traversed $n$ times is given by $w(n)=\exp(-\beta n)$. We show that the process exhibits a sharp phase transition between recurrence and transience. The critical value is determined by the branching-ruin number of the tree, which coincides with the Hausdorff dimension of the boundary of the tree under a suitable metric. We prove that the walk is almost surely transient when the branching-ruin number is greater than $1/2$, and recurrent when it is less than $1/2$. This resolves an open question posed by Kosygina.

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 paper analyzes true self-avoiding walks on infinite locally finite trees, where at each step the walk traverses a neighboring edge with probability proportional to its current weight w(n) = exp(-β n) after n prior traversals. It claims to establish a sharp phase transition for recurrence versus transience, with the critical threshold given by the branching-ruin number of the tree: the walk is a.s. transient when this number exceeds 1/2 and recurrent when it is below 1/2. The threshold is identified with the Hausdorff dimension of the tree boundary under the metric d(x,y) = exp(-c · dist(root, lca(x,y))), resolving an open question of Kosygina.

Significance. If the proofs hold, the result supplies a precise, geometrically interpretable criterion for the recurrence/transience dichotomy of this self-avoiding process on arbitrary trees rather than regular or regular-like ones. The direct comparison of escape probabilities to the branching-ruin number, together with the embedding into the boundary and the dimension identification, yields a parameter-free characterization that strengthens the link between the walk and the tree's metric geometry.

minor comments (3)
  1. [Introduction] The model definition in the introduction should explicitly state the range of β for which the phase transition holds (e.g., whether β > 0 is arbitrary or fixed), since the weight function depends on β yet the critical value 1/2 does not.
  2. [Proof of transience] The recursive estimates on weighted edges (used for both transience and recurrence) would benefit from a short appendix or subsection collecting the uniform constants that appear when passing from finite to infinite trees, to make the error controls fully transparent.
  3. [Main results] Notation for the branching-ruin number should be introduced once with a self-contained definition before its first use in the statement of the main theorem, rather than relying on the later identification with Hausdorff dimension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report contains no major comments, so we have no specific points requiring rebuttal or revision at this stage. We will incorporate any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes the phase transition via direct comparison of the walk's escape probability and effective resistance to the branching-ruin number, using recursive estimates on edge weights and an embedding of the tree into its boundary. The identification of the branching-ruin number with Hausdorff dimension under the exponential metric follows from standard dimension theory on trees and is not derived from the walk dynamics or any fitted parameter. No self-citations appear in the load-bearing steps, and the critical threshold of 1/2 is fixed by the geometric equivalence rather than by construction from the model itself. The proofs remain independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard Markov chain recurrence definitions, the exponential weight rule, and the branching-ruin number concept from prior tree literature.

axioms (2)
  • domain assumption The underlying graph is an infinite locally finite tree.
    Stated explicitly as the setting in which the walk is defined and the phase transition is proved.
  • domain assumption The weight of an edge after n traversals is w(n) = exp(-β n) for fixed β > 0.
    The model definition used to bias the walk away from previously traversed edges.

pith-pipeline@v0.9.0 · 10812 in / 1361 out tokens · 130246 ms · 2026-05-08T01:55:12.268244+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    Daniel J. Amit, G. Parisi, and L. Peliti. Asymptotic behavior of the “true” self-avoiding walk.Phys. Rev. B (3), 27(3):1635–1645, 1983

    work page 1983

  2. [2]

    Workshop report: Stochastic reinforcement processes

    Omer Angel, Roland Bauerschmidt, Mark Holmes, and Silke Rolles. Workshop report: Stochastic reinforcement processes. Banff International Research Station for Mathematical Innovation and Discovery, 2025

    work page 2025

  3. [3]

    Barbier-Chebbah, O

    A. Barbier-Chebbah, O. B´ enichou, and R. Voituriez. Self-interacting random walks: Aging, exploration, and first-passage times.Phys. Rev. X, 12:011052, Mar 2022

    work page 2022

  4. [4]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regular variation, volume 27 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1987

    work page 1987

  5. [5]

    Quantum walks with tuneable self-avoidance in one dimension.Scientific reports, 4(1):4791, 2014

    Elizabeth Camilleri, Peter P Rohde, and Jason Twamley. Quantum walks with tuneable self-avoidance in one dimension.Scientific reports, 4(1):4791, 2014

    work page 2014

  6. [6]

    The branching-ruin number as critical parameter of random processes on trees.Electron

    Andrea Collevecchio, Cong Bang Huynh, and Daniel Kious. The branching-ruin number as critical parameter of random processes on trees.Electron. J. Probab., 24:Paper No. 121, 29, 2019

    work page 2019

  7. [7]

    The branching-ruin number and the critical param- eter of once-reinforced random walk on trees.Comm

    Andrea Collevecchio, Daniel Kious, and Vladas Sidoravicius. The branching-ruin number and the critical param- eter of once-reinforced random walk on trees.Comm. Pure Appl. Math., 73(1):210–236, 2020

    work page 2020

  8. [8]

    Non-homogeneous random processes on trees with polynomial growth.In preparation, 2026

    Andrea Collevecchio, Tuan-Minh Nguyen, and Jack Thatcher. Non-homogeneous random processes on trees with polynomial growth.In preparation, 2026

    work page 2026

  9. [9]

    How information prospection facilitates spatial coverage of self-avoiding walks.Journal of Statistical Mechanics: Theory and Experiment, 2021(10):103212, oct 2021

    Javier Crist´ ın, Vi¸ cenc M´ endez, and Daniel Campos. How information prospection facilitates spatial coverage of self-avoiding walks.Journal of Statistical Mechanics: Theory and Experiment, 2021(10):103212, oct 2021

    work page 2021

  10. [10]

    Reinforced random walk.Probab

    Burgess Davis. Reinforced random walk.Probab. Theory Related Fields, 84(2):203–229, 1990

    work page 1990

  11. [11]

    Green function for an asymptotically stable random walk in a half space.J

    Denis Denisov and Vitali Wachtel. Green function for an asymptotically stable random walk in a half space.J. Theoret. Probab., 37(2):1745–1786, 2024

    work page 2024

  12. [12]

    Cell migration guided by long-lived spatial memory.Nature Communications, 12(1):4118, 2021

    Joseph d’Alessandro, Alex Barbier-Chebbah, Victor Cellerin, Olivier Benichou, Ren´ e Marc M` ege, Rapha¨ el Voituriez, and Benoit Ladoux. Cell migration guided by long-lived spatial memory.Nature Communications, 12(1):4118, 2021

    work page 2021

  13. [13]

    Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers ind≥3.Probab

    Ill´ es Horv´ ath, B´ alint T´ oth, and B´ alint Vet˝ o. Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers ind≥3.Probab. Theory Related Fields, 153(3-4):691–726, 2012. 44 TUAN-MINH NGUYEN

    work page 2012

  14. [14]

    Kesten, M

    H. Kesten, M. V. Kozlov, and F. Spitzer. A limit law for random walk in a random environment.Compositio Math., 30:145–168, 1975

    work page 1975

  15. [15]

    Network exploration using true self-avoiding walks.Phys

    Yup Kim, Seokjong Park, and Soon-Hyung Yook. Network exploration using true self-avoiding walks.Phys. Rev. E, 94:042309, Oct 2016

    work page 2016

  16. [16]

    true” self-avoiding walks to the T´ oth-Werner “true

    Elena Kosygina and Jonathon Peterson. Convergence of rescaled “true” self-avoiding walks to the T´ oth-Werner “true” self-repelling motion.Electron. J. Probab., 31:Paper No. 32, 41, 2026

    work page 2026

  17. [17]

    Once-excited random walks on general trees

    Duy-Bao Le and Tuan-Minh Nguyen. Once-excited random walks on general trees.arXiv preprint arXiv:2602.16934, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  18. [18]

    The Ising model and percolation on trees and tree-like graphs.Comm

    Russell Lyons. The Ising model and percolation on trees and tree-like graphs.Comm. Math. Phys., 125(2):337–353, 1989

    work page 1989

  19. [19]

    Random walk in a random environment and first-passage percolation on trees

    Russell Lyons and Robin Pemantle. Random walk in a random environment and first-passage percolation on trees. Ann. Probab., 20(1):125–136, 1992

    work page 1992

  20. [20]

    Cambridge University Press, New York, 2016

    Russell Lyons and Yuval Peres.Probability on trees and networks, volume 42 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York, 2016

    work page 2016

  21. [21]

    A. C. Maggs. Event-chain monte carlo and the true self-avoiding walk.Phys. Rev. E, 111:054104, May 2025

    work page 2025

  22. [22]

    Cambridge University Press, Cambridge, 2017

    Mikhail Menshikov, Serguei Popov, and Andrew Wade.Non-homogeneous random walks, volume 209 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2017. Lyapunov function methods for near-critical stochastic systems

    work page 2017

  23. [23]

    S. P. Obukhov and L. Peliti. Renormalisation of the “true” self-avoiding walk.J. Phys. A, 16(5):L147–L151, 1983

    work page 1983

  24. [24]

    Peliti and L

    L. Peliti and L. Pietronero. Random walks with memory.Riv. Nuovo Cim., 10(6):1–33, 1987

    work page 1987

  25. [25]

    Anomalous diffusion and run-and-tumble motion of a chemotactic particle in low dimensions.Phys

    Jacopo Romano and Andrea Gambassi. Anomalous diffusion and run-and-tumble motion of a chemotactic particle in low dimensions.Phys. Rev. Lett., 136:107102, Mar 2026

    work page 2026

  26. [26]

    The “true” self-avoiding walk with bond repulsion onZ: limit theorems.Ann

    B´ alint T´ oth. The “true” self-avoiding walk with bond repulsion onZ: limit theorems.Ann. Probab., 23(4):1523– 1556, 1995

    work page 1995

  27. [27]

    The true self-repelling motion.Probab

    B´ alint T´ oth and Wendelin Werner. The true self-repelling motion.Probab. Theory Related Fields, 111(3):375–452, 1998

    work page 1998

  28. [28]

    One dimensional lattice random walks with absorption at a point/on a half line.J

    Kˆ ohei Uchiyama. One dimensional lattice random walks with absorption at a point/on a half line.J. Math. Soc. Japan, 63(2):675–713, 2011. (T.-M. N.)School of Mathematical Sciences, Monash University, 9 Rainforest W alk, Clayton 3800, Victoria, Australia Email address:tuanminh.nguyen@monash.edu

    work page 2011