Capacity of loop-erased random walk

Maarten Markering

arxiv: 2411.13505 · v3 · submitted 2024-11-20 · 🧮 math.PR

Capacity of loop-erased random walk

Maarten Markering This is my paper

Pith reviewed 2026-05-23 17:40 UTC · model grok-4.3

classification 🧮 math.PR

keywords loop-erased random walkcapacitystrong law of large numbersscaling limitsnon-intersection probabilitiesgrowth exponentergodicityrandom walk

0 comments

The pith

The capacity of loop-erased random walk obeys a strong law of large numbers in four and higher dimensions with an explicit limit expressed via non-intersection probabilities.

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

This paper determines the asymptotic growth of the capacity of loop-erased random walk paths of length n on the d-dimensional lattice. For dimensions four and above it proves that the capacity divided by n converges almost surely to a positive constant that is given explicitly by non-intersection probabilities between a simple random walk and a two-sided loop-erased random walk. It further establishes ergodicity of the four-dimensional loop-erased random walk. In three dimensions the capacity grows like n raised to the power one over the growth exponent and the properly scaled capacity converges in distribution to a random variable equal to the capacity of the scaling limit of three-dimensional loop-erased random walk, which also yields a capacity-parametrized scaling limit for the walk itself.

Core claim

For d greater than or equal to four the capacity of the first n steps of loop-erased random walk divided by n converges almost surely to an explicit constant constructed from the probability that a simple random walk does not intersect a two-sided loop-erased random walk; the four-dimensional case is ergodic. For d equal to three the capacity of n steps is of order n to the power one over beta with beta the growth exponent and the scaled capacity converges in law to the capacity of the scaling limit of three-dimensional loop-erased random walk, implying that the loop-erased random walk admits a scaling limit when reparametrized by capacity.

What carries the argument

The capacity of the finite loop-erased random walk path, linked in high dimensions to non-intersection probabilities of simple random walk with two-sided loop-erased random walk and in three dimensions to the capacity of the scaling limit of the loop-erased random walk.

If this is right

The capacity normalized by length converges almost surely to a deterministic positive number in dimensions four and higher.
Four-dimensional loop-erased random walk is ergodic with respect to shifts along the path.
In three dimensions the capacity grows as n to the power one over the growth exponent of the loop-erased random walk.
The scaled capacity in three dimensions converges in distribution to a random positive limit given by the capacity of the scaling limit.
The loop-erased random walk in three dimensions possesses a scaling limit when parametrized by its capacity instead of step number.

Where Pith is reading between the lines

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

Direct numerical verification of the explicit constant could be performed by simulating long loop-erased paths and measuring their capacities in four dimensions.
The random limit object in three dimensions may allow computation of other path functionals such as length or diameter through the same scaling.
Ergodicity might be used to obtain laws of large numbers for additional additive functionals of the four-dimensional path.
The non-intersection probability expressions could connect the capacity to other geometric properties of random walk paths.

Load-bearing premise

The results in three dimensions depend on the existence of a scaling limit for three-dimensional loop-erased random walk together with its capacity scaling according to the growth exponent.

What would settle it

Numerical evidence that the capacity of n-step loop-erased random walk in four dimensions fails to approach a constant multiple of n, or that the distribution of the scaled capacity in three dimensions does not stabilize to a non-degenerate random variable.

read the original abstract

We study the capacity of loop-erased random walk (LERW) on $\mathbb{Z}^d$. For $d\geq4$, we prove a strong law of large numbers and give explicit expressions for the limit in terms of the non-intersection probabilities of a simple random walk and a two-sided LERW. Along the way, we show that four-dimensional LERW is ergodic. For $d=3$, we show that the scaling limit of the capacity of LERW is random. We show that the capacity of the first $n$ steps of LERW is of order $n^{1/\beta}$, with $\beta$ the growth exponent of three-dimensional LERW. We express the scaling limit of the capacity of LERW in terms of the capacity of Kozma's scaling limit of LERW. As a corollary, we obtain the scaling limit of the LERW in three dimensions when parametrized by its capacity.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit strong-law limits for LERW capacity in d≥4 and a random scaling limit in d=3 via Kozma's object plus a capacity parametrization corollary.

read the letter

The paper establishes a strong law for the capacity of loop-erased random walk in dimensions 4 and higher, with explicit limit expressions built from non-intersection probabilities of simple random walk and two-sided LERW. It also shows ergodicity for the four-dimensional case. In three dimensions it proves that capacity of the first n steps grows like n to the power 1/β and that the scaling limit equals the capacity of Kozma's scaling-limit object, which yields the corollary that LERW itself has a scaling limit when parametrized by capacity. These explicit formulas and the capacity-based reparametrization are new relative to the cited literature. The high-dimensional results appear to rest on standard intersection-probability calculations and look technically grounded. The d=3 claims inherit their order and their random limit directly from the existence and capacity properties of Kozma's limit together with the value of the growth exponent β. If the convergence topology used for that limit does not control capacity (or if positivity and finiteness are not already settled), both the order statement and the limit identification would fail. The abstract invokes these ingredients without extra error bounds or independent checks, so the d=3 argument carries that dependence. The work is aimed at specialists already comfortable with LERW scaling limits and SLE-type objects. Such a reader would get concrete formulas and a useful reparametrization out of it. The paper is coherent enough on its own terms and connected to prior results to deserve a serious referee, mainly to verify the derivations in d≥4 and to assess how much the d=3 claims rest on unverified capacity properties of the cited limit object.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the capacity of loop-erased random walk (LERW) on the integer lattice Z^d. For dimensions d ≥ 4, it establishes a strong law of large numbers for the capacity of the initial n steps of LERW, providing explicit formulas for the limiting constant in terms of non-intersection probabilities between simple random walks and two-sided LERW; it also proves ergodicity of four-dimensional LERW. In three dimensions, the paper shows that the capacity scales as order n^{1/β} with β the growth exponent of 3D LERW, that the scaling limit is a random variable given by the capacity of Kozma's scaling limit of LERW, and derives as a corollary the scaling limit of LERW when parametrized by capacity.

Significance. If the results are established, the work makes a notable contribution to the study of LERW by providing precise asymptotic information on its capacity. The explicit, parameter-free expressions for the limit in d ≥ 4, derived from non-intersection probabilities, represent a strength, as do the ergodicity result in d=4 and the corollary on capacity-parametrized scaling limits in d=3. These connect LERW capacity to established objects in the field and could facilitate further analysis of LERW properties across dimensions.

major comments (1)

Abstract, final paragraph: The d=3 claims that the capacity of the first n steps is of order n^{1/β} and that the scaling limit equals the capacity of Kozma's scaling limit of LERW rest on the existence and specific capacity properties (a.s. positive and finite) of that scaling limit object together with the value of the growth exponent β. The manuscript invokes these directly without an independent derivation or check within the paper; clarification is needed on whether the topology of Kozma's limit guarantees that capacity is well-defined and satisfies the required almost-sure bounds, as this is load-bearing for both the order and the expression of the random limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major comment below.

read point-by-point responses

Referee: Abstract, final paragraph: The d=3 claims that the capacity of the first n steps is of order n^{1/β} and that the scaling limit equals the capacity of Kozma's scaling limit of LERW rest on the existence and specific capacity properties (a.s. positive and finite) of that scaling limit object together with the value of the growth exponent β. The manuscript invokes these directly without an independent derivation or check within the paper; clarification is needed on whether the topology of Kozma's limit guarantees that capacity is well-defined and satisfies the required almost-sure bounds, as this is load-bearing for both the order and the expression of the random limit.

Authors: We thank the referee for this observation. The d=3 results rely on the scaling limit of LERW established by Kozma. Kozma's theorem provides the limit in a topology (uniform convergence on compact sets after suitable reparametrization) in which the capacity functional is continuous for the relevant compact sets. Consequently, the capacity of the finite-n LERW paths converges to the capacity of the limit object. Almost-sure positivity of this capacity follows from the non-degeneracy of the limit (it is a non-trivial curve, not a point), while finiteness follows from compactness of the limit in R^3. The growth exponent β is the almost-sure limit whose existence is part of the same scaling theory. These are direct consequences of the cited scaling-limit theorem rather than new derivations. Nevertheless, to address the concern, we will insert a short clarifying paragraph in the revised manuscript that explicitly references the relevant statements in Kozma's work guaranteeing continuity of capacity and the a.s. bounds. This will make the dependence self-contained without altering the proofs. revision: yes

Circularity Check

0 steps flagged

No circularity detected; results expressed via independent external objects

full rationale

The paper's d≥4 results derive an SLLN with explicit limits stated in terms of non-intersection probabilities between simple random walk and two-sided LERW; these are standard, independently defined quantities not constructed inside the paper. The d=3 claims state that capacity scales as n^{1/β} and that the scaling limit equals the capacity of Kozma's prior scaling limit object; both the growth exponent β and Kozma's limit are external references, not self-derived or fitted within this work. No self-definitional equations, no fitted parameters renamed as predictions, and no load-bearing self-citations appear. The derivation chain remains self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard axioms of probability and the existence of Kozma's scaling limit; no free parameters or new entities are introduced.

axioms (2)

standard math Standard axioms of probability theory on Z^d and existence of simple random walk loop-erasure
Used to define LERW and capacity throughout.
domain assumption Existence and basic properties of Kozma's scaling limit of 3D LERW
Invoked to express the d=3 scaling limit and corollary.

pith-pipeline@v0.9.0 · 5682 in / 1395 out tokens · 44648 ms · 2026-05-23T17:40:34.222412+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.11: lim Cap_Z3(η[0,n]) / (3 n^{1/β}) = Cap_R3(γ[0,1]) in distribution, using Kozma's scaling limit and growth exponent β
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat.induction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.2 and capacity formula via Green's function; ergodicity of ˆη via Birkhoff

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

[1]

and Li, Xinyi and Shiraishi, Daisuke, journal=arXiv preprint arXiv:2403.07256, year=2024

Sharp one-point estimates and Minkowski content for the scalin g limit of three-dimensional loop-erased random walk, author=Hernandez-Torres, S. and Li, Xinyi and Shiraishi, Daisuke, journal=arXiv preprint arXiv:2403.07256, year=2024

work page arXiv 2024
[2]

Albeverio and X

S. Albeverio and X. Y. Zhou. Intersections of random walks and W iener sausages in four dimensions. Acta Appl. Math. , 45:195–237, 1996

work page 1996
[3]

Angel, D

O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraish i. Scaling limits of the three- dimensional uniform spanning tree and associated random walk. Ann. Probab., 49(6):3032 – 3105, 2021

work page 2021
[4]

Archer, A

E. Archer, A. Nachmias, and M. Shalev. The GHP scaling limit of unifo rm spanning trees in high dimensions. Communications in Mathematical Physics , 405(3):73, 2024

work page 2024
[5]

Archer and M

E. Archer and M. Shalev. The GHP scaling limit of uniform spanning tr ees of dense graphs. Random Structures & Algorithms , 65(1):149–190, 2024

work page 2024
[6]

A. Asselah. Private communication, 2023

work page 2023
[7]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the range of r andom walk on Zd. Transac- tions of the American Mathematical Society , 370(11):7627–7645, 2018

work page 2018
[8]

Asselah, B

S. Asselah, B. Schapira, and P. Sousi. Capacity of the range of t he random walk on Z4. Ann. Probab., 47(3):1447–1497, 2019

work page 2019
[9]

Y. Chang. Two observations on the capacity of the range of simp le random walks on Z3 and Z4. Electron. Commun. Probab., 25:1–9, 2017. 21

work page 2017
[10]

U. Einmahl. Extensions of results of Koml´ os, Major, and Tusn´ ady to the multivariate case. J. Multivar. Anal. , 28(1):20–68, 1989

work page 1989
[11]

R. Gray. Probability, random processes, and ergodic properties . Springer Science & Business Media, 2009

work page 2009
[12]

Halberstam and T

N. Halberstam and T. Hutchcroft. Logarithmic corrections to the Alexander-Orbach conjecture for the four-dimensional uniform spanning tree. arXiv:2211.01307, 2022

work page arXiv 2022
[13]

Hutchcroft

T. Hutchcroft. Universality of high-dimensional spanning fore sts and sandpiles. Probab. Theory Relat. Fields, 176(1):533–597, 2020

work page 2020
[14]

Hutchcroft and P

T. Hutchcroft and P. Sousi. Logarithmic corrections to scaling in the four-dimensional uniform spanning tree. 401:2115–2191, 2024

work page 2024
[15]

Jain and S

N. Jain and S. Orey. On the range of random walk. Israel J. Math. , 6:373–380, 1968

work page 1968
[16]

G. Kozma. The scaling limit of loop-erased random walk in three dime nsions. Acta Math. , 199(1):29–152, 2007

work page 2007
[17]

G. Lawler. A self-avoiding random walk. Duke Math. J. , 47(3):655–693, 1980

work page 1980
[18]

G. Lawler. Intersections of random walks . Birkh¨ auser Boston, 1991

work page 1991
[19]

Lawler and V

G. Lawler and V. Limic. Random walk: a modern introduction , volume 123. Cambridge University Press, 2010

work page 2010
[20]

Lawler, X

G. Lawler, X. Sun, and W. X. Four-dimensional loop-erased ran dom walk. Ann. Probab. , 47(6):3866–3910, 2019

work page 2019
[21]

Losev and S

I. Losev and S. Smirnov. How long are the arms in DBM? arXiv:2307.14931, 2023

work page arXiv 2023
[22]

Michaeli, A

P. Michaeli, A. Nachmias, and M. Shalev. The diameter of uniform s panning trees in high dimensions. Probab.Theory Relat. Fields , 179:261–294, 2021

work page 2021
[23]

M¨ orters and Y

P. M¨ orters and Y. Peres. Brownian motion , volume 30. Cambridge University Press, 2010

work page 2010
[24]

Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs

Y. Peres and D. Revelle. Scaling limits of the uniform spanning tree and loop-erased random walk on ﬁnite graphs. arXiv preprint math/0410430 , 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004
[25]

Sapozhnikov and D

A. Sapozhnikov and D. Shiraishi. On Brownian motion, simple paths , and loops. Probability Theory and Related Fields , 172:615–662, 2018

work page 2018
[26]

Schweinsberg

J. Schweinsberg. The loop-erased random walk and the uniform spanning tree on the four- dimensional discrete torus. Probability Theory and Related Fields , 144(3):319–370, 2009

work page 2009
[27]

Shiraishi

D. Shiraishi. Growth exponent for loop-erased random walk in th ree dimensions. Ann. Probab., 46(2):687–774, 2018. 22

work page 2018

[1] [1]

and Li, Xinyi and Shiraishi, Daisuke, journal=arXiv preprint arXiv:2403.07256, year=2024

Sharp one-point estimates and Minkowski content for the scalin g limit of three-dimensional loop-erased random walk, author=Hernandez-Torres, S. and Li, Xinyi and Shiraishi, Daisuke, journal=arXiv preprint arXiv:2403.07256, year=2024

work page arXiv 2024

[2] [2]

Albeverio and X

S. Albeverio and X. Y. Zhou. Intersections of random walks and W iener sausages in four dimensions. Acta Appl. Math. , 45:195–237, 1996

work page 1996

[3] [3]

Angel, D

O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraish i. Scaling limits of the three- dimensional uniform spanning tree and associated random walk. Ann. Probab., 49(6):3032 – 3105, 2021

work page 2021

[4] [4]

Archer, A

E. Archer, A. Nachmias, and M. Shalev. The GHP scaling limit of unifo rm spanning trees in high dimensions. Communications in Mathematical Physics , 405(3):73, 2024

work page 2024

[5] [5]

Archer and M

E. Archer and M. Shalev. The GHP scaling limit of uniform spanning tr ees of dense graphs. Random Structures & Algorithms , 65(1):149–190, 2024

work page 2024

[6] [6]

A. Asselah. Private communication, 2023

work page 2023

[7] [7]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the range of r andom walk on Zd. Transac- tions of the American Mathematical Society , 370(11):7627–7645, 2018

work page 2018

[8] [8]

Asselah, B

S. Asselah, B. Schapira, and P. Sousi. Capacity of the range of t he random walk on Z4. Ann. Probab., 47(3):1447–1497, 2019

work page 2019

[9] [9]

Y. Chang. Two observations on the capacity of the range of simp le random walks on Z3 and Z4. Electron. Commun. Probab., 25:1–9, 2017. 21

work page 2017

[10] [10]

U. Einmahl. Extensions of results of Koml´ os, Major, and Tusn´ ady to the multivariate case. J. Multivar. Anal. , 28(1):20–68, 1989

work page 1989

[11] [11]

R. Gray. Probability, random processes, and ergodic properties . Springer Science & Business Media, 2009

work page 2009

[12] [12]

Halberstam and T

N. Halberstam and T. Hutchcroft. Logarithmic corrections to the Alexander-Orbach conjecture for the four-dimensional uniform spanning tree. arXiv:2211.01307, 2022

work page arXiv 2022

[13] [13]

Hutchcroft

T. Hutchcroft. Universality of high-dimensional spanning fore sts and sandpiles. Probab. Theory Relat. Fields, 176(1):533–597, 2020

work page 2020

[14] [14]

Hutchcroft and P

T. Hutchcroft and P. Sousi. Logarithmic corrections to scaling in the four-dimensional uniform spanning tree. 401:2115–2191, 2024

work page 2024

[15] [15]

Jain and S

N. Jain and S. Orey. On the range of random walk. Israel J. Math. , 6:373–380, 1968

work page 1968

[16] [16]

G. Kozma. The scaling limit of loop-erased random walk in three dime nsions. Acta Math. , 199(1):29–152, 2007

work page 2007

[17] [17]

G. Lawler. A self-avoiding random walk. Duke Math. J. , 47(3):655–693, 1980

work page 1980

[18] [18]

G. Lawler. Intersections of random walks . Birkh¨ auser Boston, 1991

work page 1991

[19] [19]

Lawler and V

G. Lawler and V. Limic. Random walk: a modern introduction , volume 123. Cambridge University Press, 2010

work page 2010

[20] [20]

Lawler, X

G. Lawler, X. Sun, and W. X. Four-dimensional loop-erased ran dom walk. Ann. Probab. , 47(6):3866–3910, 2019

work page 2019

[21] [21]

Losev and S

I. Losev and S. Smirnov. How long are the arms in DBM? arXiv:2307.14931, 2023

work page arXiv 2023

[22] [22]

Michaeli, A

P. Michaeli, A. Nachmias, and M. Shalev. The diameter of uniform s panning trees in high dimensions. Probab.Theory Relat. Fields , 179:261–294, 2021

work page 2021

[23] [23]

M¨ orters and Y

P. M¨ orters and Y. Peres. Brownian motion , volume 30. Cambridge University Press, 2010

work page 2010

[24] [24]

Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs

Y. Peres and D. Revelle. Scaling limits of the uniform spanning tree and loop-erased random walk on ﬁnite graphs. arXiv preprint math/0410430 , 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004

[25] [25]

Sapozhnikov and D

A. Sapozhnikov and D. Shiraishi. On Brownian motion, simple paths , and loops. Probability Theory and Related Fields , 172:615–662, 2018

work page 2018

[26] [26]

Schweinsberg

J. Schweinsberg. The loop-erased random walk and the uniform spanning tree on the four- dimensional discrete torus. Probability Theory and Related Fields , 144(3):319–370, 2009

work page 2009

[27] [27]

Shiraishi

D. Shiraishi. Growth exponent for loop-erased random walk in th ree dimensions. Ann. Probab., 46(2):687–774, 2018. 22

work page 2018