Extremal properties of the random walk local time

Marek Biskup

Extremal properties of the random walk local time

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a planar simple random walk, the set of points still unvisited when the boundary clock reaches a θ-fraction of the cover time converges, after rescaling, to the same random fractal measure that describes the √θ-thick points of the…

desk verdict Well-crafted lecture notes on a major result in 2D random walk local time and DGFF thick points, but no new mathematics; review as exposition, not research. read the letter →

arxiv 2502.09853 v1 pith:G4Q7Q7T4 submitted 2025-02-14 math.PR

Marek Biskup This is my paper · ORCID

classification math.PR MSC 60J5560G6060F0560J65

keywords randomwalklocaltimeavoidedpointscoverdiscreteGaussianfreefieldthickSecondRay-Knighttheoremtwo-dimensionalsimpleadmissibledomain

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that, for a two-dimensional simple random walk on a lattice approximation of a planar domain, the spatial distribution of the points not yet visited at a fixed fraction θ of the cover time has a non-trivial scaling limit. Time is measured by the local time accumulated at a distinguished boundary vertex rather than by the walk's own clock. In that parametrization, the normalized counting measure of avoided points converges in law to the measure $Z_D^{{√θ}}$ that describes the √θ-thick points of the discrete Gaussian free field. This connects an extremal quantity of the random walk trajectory to a canonical logarithmically correlated Gaussian field. The course lectures develop the proof through the Second Ray-Knight theorem, Kac moment formulas, and first- and second-moment estimates of the avoided-point process.

What carries the argument

The load-bearing identity is the Second Ray-Knight theorem, which couples the boundary-time-parametrized local time L_t of the walk with two independent discrete Gaussian free fields h and h̃ on the same graph through L_t + ½h² = ½(h̃ + √(2t))². This turns zero-local-time points into points where h̃ is near -√(2t), i.e., into thick points of a DGFF. Supporting machinery includes the Kac moment formula for Laplace transforms of local time, the Gibbs-Markov property of the DGFF (harmonic decomposition of the field), Green-function asymptotics with the conformal radius r_D(x), and an extended process that tracks both small local time and small field values, whose joint convergence is established by first- and second-moment estimates.

What would settle it

Simulate a simple random walk on a large square lattice approximation, run it until the boundary local time reaches t_N with θ=1/2, and record the avoided points; if the normalized empirical measure does not converge (along a sequence of squares) to a diffuse measure with total mass matching E[$Z_D^{{1/√2}}$(D)]—or if the set fails to have Hausdorff dimension near 2(1-1/2)=1—the central equivalence with DGFF thick points would be refuted.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.7: for admissible lattice approximations D_N of an admissible planar domain D, if t_N is chosen so that θ = lim t_N/(2g(log N)^2) lies in (0,1), then the measure $N^{{-2}}$$e^{{t_N/(g log N)}}$ times the sum of point masses at vertices with zero local time at the boundary-parametrized time t_N converges in law to $Z_D^{{√θ}}$. The punchline is that avoided points at a θ-fraction of the cover time are asymptotically distributed exactly as the √θ-thick points of the DGFF, where a point is thick if the field value exceeds roughly 2√(gλ) log N. Because the normalization exponent is $N^{{2(1-θ)+o(1)}}$, the measure vanishes when θ>1, consistent with the cover-time scale. The result also yields a limit law for the cardinality of the avoided set, normalized by the same factor, converging to the total mass of $Z_D^{{√θ}}$.

Load-bearing premise

The argument collapses if the Green-function asymptotics of Theorem 2.1 fail, which requires the domain and its lattice approximations to be admissible; rough boundaries or non-admissible discretizations can break the conformal-radius term r_D(x) and the logarithmic covariance structure, so the limiting measure would not have the stated form.

Editorial extensions

If this is right

At a θ-fraction of the boundary-clock cover time, the avoided points of a 2D random walk form the same random fractal, in law, as the √θ-thick points of the DGFF; their normalized number converges to the total mass of Z_D^{√θ}.
For θ>1, the avoided set is asymptotically empty, pinpointing the leading order of the cover time in this parametrization.
The same machinery yields distributions for other exceptional level sets of the local time, including λ-thick and λ-thin points, again described by the measures Z_D^λ.
The result exhibits universality of the DGFF for extremal problems of logarithmically correlated processes in two dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct corollary not spelled out in the notes is that the fractal dimension of the avoided set should be 2(1-θ) almost surely, matching the known carrier dimension of Z_D^λ; this is a testable prediction for simulations in the boundary-time parametrization.
Under the natural (walk) time parametrization, the limit becomes a differently tilted measure built from the DGFF conditioned to have zero spatial average; this shows that the boundary-clock parametrization is not just a convenience but changes the answer.
The argument is restricted to the wired-boundary construction via a single boundary vertex; extending the same statement to free boundary conditions or the lattice torus, where the Second Ray-Knight coupling is unavailable, remains an open problem suggested by the notes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: this is not a research paper. It is an expanded minicourse presenting the author's own theorems (Biskup–Louidor on DGFF thick points, Abe–Biskup on avoided points of 2D random walk) with proof sketches, exercises, and open problems. There are no new theorems, and the paper says so outright. If you pick it up expecting new results, you'll be disappointed. But as lecture notes, it's very good.

The notes' real value is pedagogical. The route from the Second Ray–Knight coupling to the avoided-point limit is laid out coherently: set up the extended process, identify subsequential limits, factor the measure, then remove the absolutely continuous part to isolate the atom at zero. The author is honest about which steps are sketched: the proof of Theorem 1.5 is carried out only for lambda < 1/sqrt(2), and the key technical step in Lemma 4.5 is deferred to [2]. That is acceptable in this format. The figures are helpful, and the appendix gives a nice summary of related results and conjectures (frequent points, cover time).

Soft spots, in proportion: the novelty is zero by design, so the paper's worth depends entirely on the quality of the exposition, which is high. The deferred technical steps are flagged explicitly and point to the right references, so a reader can fill gaps. The admissibility conditions on the domain are restrictive but stated clearly; the proofs do not overclaim. The citation pattern is fine — the author cites his own prior work because the results are his, and also gives proper credit to Eisenbaum–Kaspi–Marcus–Rosen–Shi, Dynkin, etc.

Who is this for? Graduate students or probabilists entering the area who want a fast but reliable introduction to the thick-point/avoided-point connection. It would make a good reading group source. For a journal, the right question is whether the venue publishes expository lecture notes. If yes, this deserves a serious referee — the exposition is subtle and easy to get wrong, so an expert check is worthwhile. If the venue only takes original research, desk reject is defensible, but I would not call the paper deficient on those grounds.

I would bring it to a reading group, but I wouldn't cite it in my own work beyond a background reference; I'd cite the original papers instead.

Referee Report

0 major / 5 minor

Summary. These expanded lecture notes provide an expository account of the recent theorem of Abe and Biskup (Theorem 1.7) that, for a simple random walk on an admissible lattice approximation of a bounded planar domain, the set of points still unvisited at a θ-fraction of the cover time, parametrized by local time at the boundary vertex and normalized by N^2 e^{-t_N/(g log N)}, converges in law to the √θ-thick point measure of the discrete Gaussian free field. The notes develop the necessary background: Green function asymptotics with the conformal radius, the second Ray-Knight theorem via the Kac moment formula, the Gibbs-Markov property of the DGFF, the existence and characterization of thick-point limits, and finally the thinning argument that converts avoided points into DGFF thick points. Theorems are stated in the full generality of the original papers, while several technical steps are explicitly marked as deferred to the literature.

Significance. If the result holds—and it is an already published theorem in [2]—these notes make a technically demanding circle of ideas accessible to a graduate-level audience. The main strength is transparency: the author repeatedly and explicitly marks where a proof is sketched, where a step is omitted, and where a cited result is used (see, e.g., Section 4.3 'modulo a technical step', Section 3.2 'only carry out the proof under the assumption λ < 1/√2', and Lemma 3.8 'Proof (idea)'). This is exactly the right tone for lecture notes and makes the document a valuable companion to [7] and [12]. The paper also records open questions and conjectures (Sections 5.2–5.3) that may stimulate further research. For a research journal, the novelty is limited because the central theorems are not new, but as an expository contribution the notes are of high quality and fill a pedagogical gap.

minor comments (5)

[Title page] There is a typo in 'Alfr ´ed R ´enyi Insititute'; it should read 'Institute'.
[§2.1, Eq. (2.3)] The definition of r_D(x) via the integral over the boundary is followed by a remark that it coincides with the conformal radius for simply connected D; a one-line justification or a precise reference would help the reader appreciate why this quantity is natural.
[§4.3, Lemma 4.5] The proof is explicitly 'modulo a technical step' and refers to [2, Lemma 7.1]; since this lemma is the key bridge between the local-time measure and the extended process, the introduction to Lecture 4 should state at the outset that a complete proof of this step is deferred to [2].
[§5.3, Eq. (5.20)] The event in the probability is typeset awkwardly ('Pϱ d τcov/deg(DN) ≤ ...'); adding parentheses to make the event unambiguous would improve readability.
[References] The surname of the coauthor of [27] is misspelled as 'Fitzimmons' both in the bibliography and in the body text; it should be 'Fitzsimmons'.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: Theorem 1.7 is a cited theorem whose proof reduces to the independent DGFF thick-point limit Theorem 1.5 and the Ray-Knight coupling, with no parameter fitting or definitional equivalence.

full rationale

The paper's central claim, Theorem 1.7, is attributed to the author's joint paper [2] and then proved in Lecture 4 from Theorem 1.5, the Biskup-Louidor thick-point limit for the DGFF. The proof chain is not circular: Lemma 4.1 computes P(L_t(x)=0)=e^{-t/G(x,x)}; Corollary 4.4 gives tightness; Lemma 4.5 passes to the extended process; Lemma 4.6 uses the Second Ray-Knight coupling and Theorem 1.5 to identify subsequential limits; Corollary 4.7 and Theorem 4.8 identify the measure via the Laplace transform of the local-time component; and Lemma 4.9 removes the absolutely continuous contribution, yielding Theorem 1.7. The limiting object Z_D^{sqrt(theta)} is defined independently in Theorem 1.5 as the DGFF thick-point measure, not in terms of avoided points, and no parameter is fitted to the avoided-point data. The only noteworthy feature is heavy self-citation: the notes explicitly defer the proof of Theorem 1.5 for lambda >= 1/sqrt(2) to the author's PIMS notes [7] and the original paper [12], and defer several technical lemmas (Lemma 2.5, Lemma 3.3, Lemma 4.5 truncation) to the author's prior work. This is a normal reliance on published results, and those results do not assume the target theorem; Theorem 1.5 is a separately established benchmark, and the proof of Theorem 1.7 via Ray-Knight is a substantive derivation rather than a renaming or definitional shortcut. Accordingly, no specific circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. The measure Z_D^λ and the distribution μ in Theorem 4.8 are defined objects from prior work, not fitted quantities.

assumptions (4)

standard math Potential kernel asymptotic a(x) = g log|x| + c0 + O(|x|^{-2}) for the 2D simple random walk (Lemma 2.3).
Invoked in the proof of Green function asymptotics (Theorem 2.1). Cited to Stöhr [42] and Kozma-Schreiber [33].
domain assumption Second Ray-Knight Theorem: for each t≥0 there is a coupling of the local time L_t and two DGFFs h and h~ such that L_t(x)+h_x^2/2 = (h~_x + sqrt(2t))^2/2 (Theorem 2.7).
The core link between local time and DGFF used to prove Theorem 1.7. Attributed to Eisenbaum et al. [25] and Zhai [44].
domain assumption The limit measure Z_D^λ exists, is unique, and is given by Gaussian multiplicative chaos for the continuum GFF (used in proving Theorem 1.5).
The proof of Theorem 1.5 relies on uniqueness of the LQG measure via Kahane's criterion and the work of Berestycki-Powell [6] and others.
domain assumption Admissible domains and approximations (Definitions 1.3 and 1.4) yield uniform Green function asymptotics with conformal radius r_D(x).
This is the technical foundation for the first and second moment estimates; if D is not admissible, the logarithmic scaling may fail.

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Pith. "Pith review of Extremal properties of the random walk local time." pith.science (2026). https://pith.science/paper/G4Q7Q7T4

@misc{pith2026250209853,
  author       = {Pith},
  title        = {Pith review of: Extremal properties of the random walk local time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4Q7Q7T4}},
  note         = {Machine review of arXiv:2502.09853}
}

read the original abstract

These are expanded lecture notes for a minicourse taught at the "School on disordered media" at the Alfred Renyi institute in Budapest, January 2025.

Figures

Figures reproduced from arXiv: 2502.09853 by the authors.

**Figure 1.** An illustration of the state space for the random walk. Here DN is simply an N ˆ N square while ϱ is a vertex to which all the boundary edges of DN in Z2 are re-routed. Note that the invariant measure π(x) equals the degree of the vertex x in the resulting graph which for DN with wired boundary condition will be equal 2d at x P DN and equal to the size of the edge boundary of DN in Zd at ϱ. The chain X is then a con… view at source ↗

**Figure 2.** A sample of the local time (left) and the trajectory of the walk (right) over time (with time axis running upwards) for the random walk on DN Y tϱu as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Samples of the set of avoided points for the random walk on DN run for time proportional to θ = 0.1 (left) and θ = 0.3 (right) fraction of the expected cover time. (The same run of the random walk is used for both figures.) concentration fails in d = 1 as τcov/N2 tends in law (as N Ñ 8) to a non-degenerate random variable. The formula Nd log N in d ě 3 is easy to understand: One needs Nd time to visit most of the ve… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: A sample of DGFF on 500ˆ500-square color coded so that the red regions are those with large positive values and purple regions are those with large negative values. The values in-between are coded according to usual ordering of colors by the wave-length. Similarly as f…

**Figure 5.** Figure 5: An illustration of an admissible approximation DN (marked by the lattice side in the dark region) of an admissible domain D Ď R2 (bounded by the thick lines). also allow for non-trivial arcs in the interior. While connectedness is not required, the fact that our proces…

**Figure 6.** Figure 6: The level sets at λ := 0.1 (left) and λ = 0.3 (right) multiple of the expected maximum of DGFF on a square box of side-length 500.. We now define the set of λ-thick points again as TN(λ) := ␣ x P DN : h DN x ě 2 ? λg log N ( , λ P (0, 1). (1.25) As noted earlier, this …

**Figure 7.** Figure 7: An illustration of the geometric setting for one typical use of the Gibbs-Markov property. Here V is a box of (2N ´ 1) 2 vertices which is split into four (N ´ 1) ˆ (N ´ 1) squares (whose union is U) and a “cross” made of two lines of vertices separating these. of vert…

**Figure 8.** Figure 8: A sample of φ V,U for the geometric setting in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

**Figure 9.** Figure 9: for a sample of Z D λ -measure [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]

**Figure 10.** Figure 10: A comparison between the large values of the DGFF (left) and i.i.d. Gaussians (right) demonstrating the clustering of the DGFF values. which is similar to the processes associated with the λ-thick points, albeit without any normalization. For this process, the above p…

**Figure 11.** Figure 11: A plot of function d on D := (0, 1) ˆ (0, 1) obtained by solving the Poisson equation with constant charge density ´∆d = Leb(D)/Var(Y). then Y KK Z D,0 λ ñ e λαd(x)Y Z D,0 λ (dx) law= Z D λ (dx) (5.13) where d(x) := Leb(D) ş D dy GpD(x, y) ş DˆD dz dy GpD(z, y) (5.14)…

**Figure 12.** Figure 12: A plot of the range (left) and the local time configuration (right) of the simple random walk started at the origin of a lattice square and run until the first exit. domains with free boundary conditions and the lattice torus. These cases are currently subject of vari…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

Abe and M

Y. Abe and M. Biskup (2022). Exceptional points of two-dimensional random walks at multiples of the cover time. Probab. Theory Rel. Fields 183, 1–55

work page 2022

[2] [7]

Biskup (2020)

M. Biskup (2020). Extrema of the two-dimensional Discrete Gaussian Free Field. In: M. Barlow and G. Slade (eds.): Random Graphs, Phase Transitions, and the Gaussian Free Field. SSPROB 2017. Springer Proceedings in Mathematics & Statistics, 304 (2020) 163–407. Springer, Cham

2020

[3] [12]

Biskup and O

M. Biskup and O. Louidor (2019). On intermediate level sets of two-dimensional discrete Gaussian free field. Ann. Inst. Henri Poincar´ e55, no. 4, 1948–1987

work page 2019

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Abe (2018)

Y. Abe (2018). Extremes of local times for simple random walks on symmetric trees. Elec- tron. J. Probab. 23, paper no. 40, 41 pp

work page 2018

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Y. Abe, M. Biskup and S. Lee (2023). Exceptional points of discrete-time random walks in planar domains. Electron. J. Probab. 28, paper no. 137, 1–45

work page 2023

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D. Belius (2013). Gumbel fluctuations for cover times in the discrete torus. Probab. Theory Rel. Fields 157, 635–689

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Belius, J

D. Belius, J. Rosen, and O. Zeitouni (2020). Tightness for the cover time of the two dimen- sional sphere. Probab. Theory Rel. Fields 176, no. 3-4, 1357–1437

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Berestycki and E

N. Berestycki and E. Powell (2024). Gaussian free field and Liouville quantum gravity. arXiv:2404.1664

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