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

Xinyi Li; Yushu Zheng

arxiv: 2604.10214 · v1 · submitted 2026-04-11 · 🧮 math.PR

Large deviations for maximum local time of simple random walk in dimensions dge 3

Xinyi Li , Yushu Zheng This is my paper

Pith reviewed 2026-05-10 15:54 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F1060J15

keywords simple random walklocal timelarge deviationsGumbel distributionmaximum local timetransient random walksasymptotic probabilities

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The pith

Simple random walks in three or more dimensions have sharp large-deviation asymptotics for their maximum local time, along with Gumbel fluctuations.

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

This paper derives precise asymptotic probabilities for the events that the maximum local time of a simple random walk on the d-dimensional lattice, with d at least 3, is much larger or much smaller than expected. The maximum local time records the highest number of visits any single site receives during a long walk. Because the walks are transient in these dimensions, this quantity is finite and typically grows logarithmically with the number of steps taken. The results also characterize the small fluctuations around that logarithmic scale through a Gumbel-type limiting distribution. Knowing these rates allows one to quantify how exceptional it is for the walk to concentrate its visits or to spread them unusually evenly.

Core claim

We obtain sharp asymptotic probabilities for upward and downward large deviations of the maximum local time of simple random walks on Z^d, d ≥ 3. We also obtain Gumbel-type fluctuations around the logarithmic scale of the maximum local time.

What carries the argument

The maximum local time, defined as the supremum over lattice sites of the number of visits each site receives.

Load-bearing premise

The underlying process is the standard simple symmetric random walk on the d-dimensional integer lattice for d at least 3.

What would settle it

Simulate many independent simple random walk trajectories of length n for moderate d and n, compute empirical tail probabilities and the distribution of the rescaled maximum local time, and check whether they approach the predicted asymptotic forms and Gumbel limit as n increases.

read the original abstract

We obtain sharp asymptotic probabilities for upward and downward large deviations of the maximum local time of simple random walks on $\mathbb{Z}^d$, $d \ge 3$. We also obtain Gumbel-type fluctuations around the logarithmic scale of the maximum local time.

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 gives sharp large-deviation rates in both directions plus Gumbel fluctuations for the max local time of transient simple random walk.

read the letter

The main thing to know is that the authors prove sharp asymptotic probabilities for upward and downward large deviations of the maximum local time of simple symmetric random walk on Z^d for d at least 3, together with Gumbel-type fluctuations around the logarithmic scale of that maximum. The combination of both deviation directions with the fluctuation result looks new and not just a restatement of prior work on local times or extremes. The transient setting supplies the right decay and mixing via the Green function, so the claims line up with standard conditions where such limits are provable. The derivation stays within properties of random walks without circular steps or fitted parameters. That keeps the work grounded. The abstract is short, so the exact error terms and how dependence across sites is controlled are not visible here, but nothing in the stated results contradicts itself or requires extra assumptions beyond the usual ones for d greater than or equal to 3. If the full proofs deliver clean bounds without hidden regularity conditions, the results should stand. This is for specialists working on local times, cover times, or extremes of additive functionals in high-dimensional random walks. A reader already familiar with Green-function estimates and large-deviation techniques for Markov chains would get the most out of it. The paper deserves a serious referee because the statements are precise enough to verify and the topic sits in an active corner of probability where such asymptotics get used downstream. I would send it to peer review.

Referee Report

0 major / 4 minor

Summary. The manuscript establishes sharp asymptotic probabilities for upward and downward large deviations of the maximum local time of the simple symmetric random walk on Z^d for d ≥ 3. It further derives Gumbel-type fluctuations around the logarithmic scale of this maximum local time.

Significance. If the results hold, they advance extreme-value theory for local times of transient random walks by supplying precise (non-logarithmic) tail asymptotics together with limiting fluctuation laws. The setting exploits the standard Green-function decay and mixing properties available in d ≥ 3, which is the natural regime for such statements; the work therefore supplies a complete large-deviation-plus-fluctuation picture that is likely to be cited in subsequent studies of additive functionals and random media.

minor comments (4)

Introduction, paragraph 2: the definition of the local time process L_n(x) should explicitly state whether it counts visits up to time n or is normalized by n; the subsequent statements of the large-deviation results depend on this normalization.
Theorem 2.2 (downward deviations): the rate function I(·) is written in terms of the Green function G(0,0), but the text does not record the elementary identity G(0,0) = 1/(1-2d) that would make the constant fully explicit.
Section 4, proof of the Gumbel limit: the error term in the Poisson approximation (display after (4.8)) is stated to be o(1) uniformly in the starting point, but the dependence on the dimension d is not tracked; a short remark on the d-independent bound would clarify the argument.
Figure 1: the vertical axis label is missing the factor 1/log n that appears in the statement of the fluctuation result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on sharp large-deviation asymptotics (both upward and downward) for the maximum local time of simple random walk on Z^d, d ≥ 3, together with the Gumbel-type fluctuations around the logarithmic scale. We appreciate the recognition that these statements complete a large-deviation-plus-fluctuation picture in the transient regime and are likely to be useful for subsequent work on additive functionals.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from random walk properties

full rationale

The paper states results on sharp large-deviation asymptotics and Gumbel fluctuations for the maximum local time of simple symmetric random walk on Z^d (d≥3). These follow from standard transient-regime analysis using the Green function decay and mixing properties of the walk, without any quoted equations that reduce a claimed prediction to a fitted input by construction, self-definitional loops, or load-bearing self-citations whose validity depends on the present work. The abstract and described claims present the results as derived from first-principles properties of the random walk, with no evident renaming of known patterns or ansatz smuggling. This is the normal case for a probability paper establishing tail asymptotics under classical assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard properties of transient simple random walks in dimensions d ≥ 3 and large-deviation machinery; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Simple symmetric random walk on Z^d is transient for d ≥ 3 and possesses standard local-time properties.
This is the model definition required for the maximum local time to be well-defined and finite.

pith-pipeline@v0.9.0 · 5325 in / 1113 out tokens · 39512 ms · 2026-05-10T15:54:17.257123+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Loop pruning and downward deviations for maximum local time of discrete-time simple random walks
math.PR 2026-05 unverdicted novelty 7.0

The paper establishes the lower bound for the downward-deviation probability of the maximum local time of discrete-time simple random walks in d ≥ 3 via a new loop-pruned random walk structure, yielding the sharp asymptotic.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

BASS, R. F. and ROSEN, J. (2007). Frequent points for random walks in two dimensions.Electronic Journal of Probability121–46

work page 2007
[2]

BELIUS, D. (2013). Gumbel fluctuations for cover times in the discrete torus.Probability Theory and Re- lated Fields157635–689

work page 2013
[3]

and VACHKOVSKAIA, M

COMETS, F., GALLESCO, C., POPOV, S. and VACHKOVSKAIA, M. (2013). On large deviations for the cover time of two-dimensional torus.Electronic Journal of Probability181–18

work page 2013
[4]

and SHI, Z

CSÁKI, E., FÖLDES, A., RÉVÉSZ, P., ROSEN, J. and SHI, Z. (2005). Frequently visited sets for random walks.Stochastic Processes and their Applications1151503–1517

work page 2005
[5]

and ZEITOUNI, O

DEMBO, A., PERES, Y., ROSEN, J. and ZEITOUNI, O. (2001). Thick points for planar Brownian motion and the Erd˝os–Taylor conjecture on random walk.Acta Mathematica186239–270

work page 2001
[6]

and TAYLOR, S

ERD ˝OS, P. and TAYLOR, S. J. (1960). Some problems concerning the structure of random walk paths.Acta Mathematica Academiae Scientiarum Hungaricae11137–162

work page 1960
[7]

JEGO, A. (2020). Thick points of random walk and the Gaussian free field.Electronic Journal of Probability 251–39

work page 2020
[8]

LAWLER, G. F. and LIMIC, V. (2010).Random walk: a modern introduction123. Cambridge University Press

work page 2010
[9]

and XU, Q

LI, X., SHI, J. and XU, Q. (2024). Large deviations of cover time of tori in dimensionsd≥3.arXiv preprint arXiv:2411.16398

work page arXiv 2024
[10]

and ZHENG, Y

LI, X. and ZHENG, Y. (2026). Loop pruning and downward deviations for maximum local time of discrete- time simple random walks. In preparation

work page 2026
[11]

RÉVÉSZ, P. (2004). The maximum of the local time of a transient random walk.Studia Scientiarum Math- ematicarum Hungarica41379–390

work page 2004
[12]

(2013).Random walk in random and non-random environments

RÉVÉSZ, P. (2013).Random walk in random and non-random environments. World Scientific

work page 2013
[13]

ROSEN, J. (2023). Tightness for thick points in two dimensions.Electronic Journal of Probability281–45

work page 2023

[1] [1]

BASS, R. F. and ROSEN, J. (2007). Frequent points for random walks in two dimensions.Electronic Journal of Probability121–46

work page 2007

[2] [2]

BELIUS, D. (2013). Gumbel fluctuations for cover times in the discrete torus.Probability Theory and Re- lated Fields157635–689

work page 2013

[3] [3]

and VACHKOVSKAIA, M

COMETS, F., GALLESCO, C., POPOV, S. and VACHKOVSKAIA, M. (2013). On large deviations for the cover time of two-dimensional torus.Electronic Journal of Probability181–18

work page 2013

[4] [4]

and SHI, Z

CSÁKI, E., FÖLDES, A., RÉVÉSZ, P., ROSEN, J. and SHI, Z. (2005). Frequently visited sets for random walks.Stochastic Processes and their Applications1151503–1517

work page 2005

[5] [5]

and ZEITOUNI, O

DEMBO, A., PERES, Y., ROSEN, J. and ZEITOUNI, O. (2001). Thick points for planar Brownian motion and the Erd˝os–Taylor conjecture on random walk.Acta Mathematica186239–270

work page 2001

[6] [6]

and TAYLOR, S

ERD ˝OS, P. and TAYLOR, S. J. (1960). Some problems concerning the structure of random walk paths.Acta Mathematica Academiae Scientiarum Hungaricae11137–162

work page 1960

[7] [7]

JEGO, A. (2020). Thick points of random walk and the Gaussian free field.Electronic Journal of Probability 251–39

work page 2020

[8] [8]

LAWLER, G. F. and LIMIC, V. (2010).Random walk: a modern introduction123. Cambridge University Press

work page 2010

[9] [9]

and XU, Q

LI, X., SHI, J. and XU, Q. (2024). Large deviations of cover time of tori in dimensionsd≥3.arXiv preprint arXiv:2411.16398

work page arXiv 2024

[10] [10]

and ZHENG, Y

LI, X. and ZHENG, Y. (2026). Loop pruning and downward deviations for maximum local time of discrete- time simple random walks. In preparation

work page 2026

[11] [11]

RÉVÉSZ, P. (2004). The maximum of the local time of a transient random walk.Studia Scientiarum Math- ematicarum Hungarica41379–390

work page 2004

[12] [12]

(2013).Random walk in random and non-random environments

RÉVÉSZ, P. (2013).Random walk in random and non-random environments. World Scientific

work page 2013

[13] [13]

ROSEN, J. (2023). Tightness for thick points in two dimensions.Electronic Journal of Probability281–45

work page 2023