Loop pruning and downward deviations for maximum local time of discrete-time simple random walks

Xinyi Li; Yushu Zheng

arxiv: 2605.16086 · v1 · pith:PGXIV3IWnew · submitted 2026-05-15 · 🧮 math.PR

Loop pruning and downward deviations for maximum local time of discrete-time simple random walks

Xinyi Li , Yushu Zheng This is my paper

Pith reviewed 2026-05-19 19:11 UTC · model grok-4.3

classification 🧮 math.PR

keywords random walklocal timemaximum local timedownward deviationlarge deviationloop pruningdiscrete timeinteger lattice

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The pith

Loop pruning transfers the continuous-time lower bound to prove sharp asymptotics for downward deviations of discrete random walk local times.

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

The paper proves the lower bound for the probability of large downward deviations in the maximum local time of discrete-time simple random walks on Z^d with d at least 3. This completes the sharp asymptotic formula, since the upper bound was already known from prior work. The authors achieve this by introducing the loop-pruned random walk and its decomposition, which serves as a discrete analogue to the jump-chain and holding-time structure in continuous time. This new structure allows the proof techniques from the continuous case to be adapted with controlled errors. A reader would care because it resolves the open matching lower bound and provides a tool that may apply to other discrete stochastic processes.

Core claim

The central discovery is that the loop-pruning decomposition of the discrete-time simple random walk path enables a direct lower bound proof for the downward deviation probability of its maximum local time, yielding the same sharp asymptotic rate as in the continuous-time setting.

What carries the argument

The loop-pruned random walk, constructed via the loop-pruning decomposition to mimic the essential features of continuous-time paths for deviation analysis.

If this is right

The sharp asymptotic formula for the downward-deviation probability now holds completely.
The loop-pruning technique offers a method to bridge discrete and continuous random walk analyses.
This result supports the idea that certain large deviation principles are robust across time discretizations.

Where Pith is reading between the lines

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

The decomposition might be useful for analyzing other path functionals like the range or the number of intersections.
Similar pruning ideas could be tested on random walks in lower dimensions or on different graphs.
It suggests exploring whether loop pruning can simplify proofs in related areas such as branching random walks or polymer models.

Load-bearing premise

The loop-pruning decomposition must accurately capture the path properties that matter for the deviation without introducing errors that affect the lower bound.

What would settle it

A simulation or computation of the deviation probability for large but finite n that shows whether the log-probability divided by the scaling function approaches the predicted constant from the asymptotic formula.

read the original abstract

We study downward deviations of the maximum local time of the discrete-time simple random walk on $\mathbb{Z}^d$, $d\ge 3$. In our previous paper \cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching lower bound was left open. In the present paper, we prove this lower bound and hence obtain the sharp asymptotic formula for the downward-deviation probability. To provide a discrete-time analogue of the jump-chain/holding-time structure used in the continuous-time argument, we introduce a new random structure which we name as {\it loop-pruned random walk} and the associated loop-pruning decomposition, which is also of independent interest.

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

This paper supplies the missing lower bound for downward deviations of maximum local time in discrete-time random walks by introducing loop pruning as a discrete analogue to continuous-time decompositions.

read the letter

This paper closes the open lower bound for the downward-deviation probability of the maximum local time of discrete-time simple random walks on Z^d, d at least 3. Combined with the authors' earlier upper bound, it gives the sharp asymptotic rate. The new ingredient is the loop-pruned random walk and its decomposition, which they use to carry over the continuous-time variational argument to the discrete setting. That construction is the main technical novelty and looks like it could stand on its own for other problems involving local times and excursions. The paper is straightforward about building on their prior work and states the goal clearly. The argument appears to keep the relevant path properties under control so the lower bound transfers without obvious extra exponential cost. A soft spot worth checking is whether the pruning step really keeps the Radon-Nikodym factor between the original and pruned measures from adding a non-negligible term on the event of interest. If the estimates on that factor are only logarithmic or smaller, the sharpness holds; any larger mismatch would weaken the claim. The stress-test concern about distortion in joint laws of local times and holding times is the right thing to verify in the proofs. This is a narrow but solid contribution aimed at people who follow large-deviation results for random-walk local times. A reader already working in that corner of probability theory will see the value in having the discrete case settled with an explicit new tool. It is worth sending to a serious referee because the result completes a specific open question and the new structure is reproducible enough to check.

Referee Report

2 major / 2 minor

Summary. The paper proves the matching lower bound for the downward-deviation probability of the maximum local time of discrete-time simple random walks on Z^d (d ≥ 3). Building on the authors' prior upper bound, it introduces the loop-pruned random walk and loop-pruning decomposition as a discrete-time analogue of the jump-chain/holding-time structure, thereby establishing the sharp asymptotic formula.

Significance. If the loop-pruning construction is shown to control distortions in the joint law of local times and excursion lengths without introducing an extra exponential cost, the result completes the sharp large-deviation asymptotics for the maximum local time. The new random structure may also be of independent interest for transferring continuous-time techniques to discrete settings.

major comments (2)

[Section 2] Definition of loop-pruned random walk (Section 2): the argument requires an explicit bound showing that the Radon-Nikodym derivative between the original and pruned measures has logarithm o(n) (or the appropriate scaling) on the event of interest; without this, the transfer of the continuous-time lower bound cannot be guaranteed to preserve sharpness.
[Section 4] Lower-bound proof (Section 4): the claim that the pruned process allows direct application of the variational analysis must be supported by a coupling or direct verification that the maximum local time is preserved and that conditional laws of holding times/excursion lengths introduce no uncontrolled multiplicative factor.

minor comments (2)

[Introduction] The notation distinguishing the loop-pruned walk from the original walk could be clarified with a short illustrative example or diagram early in the manuscript.
Ensure all references to the prior upper-bound paper are accompanied by precise citations to the relevant statements used here.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. The loop-pruning construction is intended to serve as a discrete-time counterpart to the jump-chain/holding-time decomposition, and we will strengthen the presentation to make the control of distortions fully explicit.

read point-by-point responses

Referee: [Section 2] Definition of loop-pruned random walk (Section 2): the argument requires an explicit bound showing that the Radon-Nikodym derivative between the original and pruned measures has logarithm o(n) (or the appropriate scaling) on the event of interest; without this, the transfer of the continuous-time lower bound cannot be guaranteed to preserve sharpness.

Authors: We agree that an explicit quantitative bound on the Radon-Nikodym derivative is needed to guarantee that the lower bound transfers without losing sharpness. In the revised manuscript we will insert a new lemma (Lemma 2.8) that bounds the logarithm of the Radon-Nikodym derivative between the law of the original walk and the law of the loop-pruned walk. The bound is obtained by estimating the total length of pruned loops on the event that the maximum local time is at most (1+ε) times its typical downward-deviation scale; standard Green-function estimates for d ≥ 3 then show that this logarithm is o(n) with probability 1−o(1) under the conditioned measure. This lemma will be placed immediately after the definition of the loop-pruned walk and will be invoked in the proof of the lower bound. revision: yes
Referee: [Section 4] Lower-bound proof (Section 4): the claim that the pruned process allows direct application of the variational analysis must be supported by a coupling or direct verification that the maximum local time is preserved and that conditional laws of holding times/excursion lengths introduce no uncontrolled multiplicative factor.

Authors: We will add a short coupling argument (Proposition 4.3) showing that the loop-pruned walk can be constructed on the same probability space as the original walk so that their local-time fields coincide exactly on the event of interest. Consequently the maximum local time is identical for both processes. We will also verify that the conditional distribution of the lengths of the pruned excursions, given the pruned skeleton, contributes only a multiplicative factor whose logarithm is absorbed into the lower-order terms of the variational problem; the leading exponential rate therefore remains unchanged. These additions will be inserted at the beginning of Section 4 before the application of the variational analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: lower bound established through independent loop-pruning construction

full rationale

The manuscript establishes the lower bound for downward deviations of the maximum local time by introducing a novel loop-pruning decomposition tailored to discrete-time random walks. This structure is defined and its properties derived within the current paper to facilitate the transfer of continuous-time techniques, without reducing the result to a rephrasing of prior inputs or self-citations. The reference to the authors' previous work applies only to the complementary upper bound, leaving the new lower-bound argument with independent mathematical content that does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the correctness of the newly defined loop-pruning decomposition and on standard background facts about transient random walks in d ≥ 3; no numerical fitting parameters are mentioned.

axioms (1)

standard math Standard properties of simple symmetric random walk on Z^d for d ≥ 3 (transience, Green function decay)
Invoked implicitly to guarantee the walk is transient so that local times are finite and maximum local time has a non-degenerate distribution.

invented entities (1)

loop-pruned random walk no independent evidence
purpose: To serve as a discrete-time analogue of the jump-chain/holding-time decomposition used in continuous-time arguments
Newly introduced in this paper to enable the lower-bound proof; no independent evidence outside the manuscript is provided.

pith-pipeline@v0.9.0 · 5639 in / 1281 out tokens · 31688 ms · 2026-05-19T19:11:29.860691+00:00 · methodology

discussion (0)

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

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Relation between the paper passage and the cited Recognition theorem.

We study downward deviations of the maximum local time of the discrete-time simple random walk on Z^d, d≥3

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Stochastic processes and their applications , volume=

Frequently visited sets for random walks , author=. Stochastic processes and their applications , volume=. 2005 , publisher=

work page 2005
[2]

2013 , publisher=

Random walk in random and non-random environments , author=. 2013 , publisher=

work page 2013
[3]

arXiv preprint arXiv:2409.00995 , year=

Favorite sites for simple random walk in two and more dimensions , author=. arXiv preprint arXiv:2409.00995 , year=

work page arXiv
[4]

Thick points of random walk and the Gaussian free field , author=

work page
[5]

Probability Theory and Related Fields , volume=

Gumbel fluctuations for cover times in the discrete torus , author=. Probability Theory and Related Fields , volume=. 2013 , publisher=

work page 2013
[6]

Large deviations of cover time of tori in dimensions d 3 , author=. Ann. Inst H. Poincar\'e, Probab. Stat. , note=

work page
[7]

Electron

On large deviations for the cover time of two-dimensional torus , author=. Electron. J. Probab , volume=

work page
[8]

2012 , publisher=

Intersections of random walks , author=. 2012 , publisher=

work page 2012
[9]

2010 , publisher=

Random walk: a modern introduction , author=. 2010 , publisher=

work page 2010
[10]

High dimensional probability V: the Luminy volume , volume=

Bernstein inequality and moderate deviations under strong mixing conditions , author=. High dimensional probability V: the Luminy volume , volume=. 2009 , publisher=

work page 2009
[11]

Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$

Large deviations for maximal local time of simple random walk in dimensions d 3 , author=. arXiv preprint arXiv:2604.10214 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[12]

1998 , publisher=

Large Deviations Techniques and Applications , author=. 1998 , publisher=

work page 1998
[13]

Some problems concerning the structure of random walk paths , journal =

Erd. Some problems concerning the structure of random walk paths , journal =. 1960 , number =

work page 1960
[14]

Studia Scientiarum Mathematicarum Hungarica , volume =

P\'al R\'ev\'esz , title =. Studia Scientiarum Mathematicarum Hungarica , volume =

work page

[1] [1]

Stochastic processes and their applications , volume=

Frequently visited sets for random walks , author=. Stochastic processes and their applications , volume=. 2005 , publisher=

work page 2005

[2] [2]

2013 , publisher=

Random walk in random and non-random environments , author=. 2013 , publisher=

work page 2013

[3] [3]

arXiv preprint arXiv:2409.00995 , year=

Favorite sites for simple random walk in two and more dimensions , author=. arXiv preprint arXiv:2409.00995 , year=

work page arXiv

[4] [4]

Thick points of random walk and the Gaussian free field , author=

work page

[5] [5]

Probability Theory and Related Fields , volume=

Gumbel fluctuations for cover times in the discrete torus , author=. Probability Theory and Related Fields , volume=. 2013 , publisher=

work page 2013

[6] [6]

Large deviations of cover time of tori in dimensions d 3 , author=. Ann. Inst H. Poincar\'e, Probab. Stat. , note=

work page

[7] [7]

Electron

On large deviations for the cover time of two-dimensional torus , author=. Electron. J. Probab , volume=

work page

[8] [8]

2012 , publisher=

Intersections of random walks , author=. 2012 , publisher=

work page 2012

[9] [9]

2010 , publisher=

Random walk: a modern introduction , author=. 2010 , publisher=

work page 2010

[10] [10]

High dimensional probability V: the Luminy volume , volume=

Bernstein inequality and moderate deviations under strong mixing conditions , author=. High dimensional probability V: the Luminy volume , volume=. 2009 , publisher=

work page 2009

[11] [11]

Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$

Large deviations for maximal local time of simple random walk in dimensions d 3 , author=. arXiv preprint arXiv:2604.10214 , year=

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

1998 , publisher=

Large Deviations Techniques and Applications , author=. 1998 , publisher=

work page 1998

[13] [13]

Some problems concerning the structure of random walk paths , journal =

Erd. Some problems concerning the structure of random walk paths , journal =. 1960 , number =

work page 1960

[14] [14]

Studia Scientiarum Mathematicarum Hungarica , volume =

P\'al R\'ev\'esz , title =. Studia Scientiarum Mathematicarum Hungarica , volume =

work page