on limit laws for two-dimensional torus cover times

Overview

The final stage of covering a two-dimensional torus is controlled by a few exceptionally persistent holes. Two October 2026 preprints identify the limiting fluctuations of this covering time: Yechi Zhou treats simple random walk on the discrete torus, while Amir Dembo, Jay Rosen, and Ofer Zeitouni treat small-radius covering by Brownian motion on the continuous torus (Zhou, 2026; Dembo, Rosen & Zeitouni, 2026). Both limits are randomly shifted Gumbel distributions. Their proofs explain how randomness accumulated across spatial scales controls the number of last surviving holes.

The papers share an extreme-value mechanism, but their technical routes differ substantially. Zhou transfers Gaussian information to local-time statistics and then to unvisited components. Dembo–Rosen–Zeitouni analyze excursions directly, obtaining precise probabilities that small blocks remain uncovered. This post develops the preliminaries and the main reductions, leaving the long uniform estimates to the papers.

This account concerns the v1 manuscripts, dated October 1 and 2, respectively. The Brownian paper’s opening statement says that its argument was developed over eight years and that Zhou’s posting prompted release of the current version. The mathematical comparison below concerns the stated results and proof architecture; it is not an independent verification of every estimate in these lengthy preprints.

🏷️ Models and Probability Preliminaries

Discrete and continuous covering

Write

$$\begin{array}{c}{V}_N=(\mathbb{Z}/N\mathbb{Z}{)}^2,\\ {\mathbb{T}}^2={\mathbb{R}}^2/{\mathbb{Z}}^2.\end{array}$$

The second torus has area one. On $V_N$, let $X_t$ be continuous-time simple random walk with total jump rate one: it waits an exponential time of mean one, then chooses one of its four neighbors uniformly. Its generator is

$$\begin{array}{c}{\mathcal{L}}_{N}f(x)\\ =\frac{1}{4}\sum _{e\in \{\pm e_1,\pm e_2\}}(f(x+e)-f(x)).\end{array}$$

Write ${\mathcal{R}}_t=\{X_s:0\le s\le t\}$ for the visited set. The cover time and local times are

$$\begin{array}{rl}{T}_N& =\inf \{t:{\mathcal{R}}_t=V_N\},\\ {\ell }_t(x)& ={\int }_0^t{1}_{\{X_s=x\}}ds.\end{array}$$

At a deterministic positive time, the unvisited vertices are almost surely exactly those with zero local time. Small positive local times describe vertices that have barely been explored.

For standard Brownian motion $B_t$ on ${\mathbb{T}}^2$, with generator $\frac{1}{2}\Delta$, define

$$\begin{array}{c}{\mathcal{C}}_{\epsilon }\\ =\inf \left\{t:\underset{x\in {\mathbb{T}}^2}{\sup }\underset{0\le s\le t}{\inf }d(x,B_s)\le \epsilon \right\}.\end{array}$$

Thus every point must come within distance $\epsilon$ of the path. Covering neighborhoods is essential: hitting every individual point is not the continuum analogue of visiting all vertices.

The first-order results of Dembo–Peres–Rosen–Zeitouni give

$$\begin{array}{c}\frac{T_N}{N^2(\log N{)}^2}⟶\frac{4}{\pi },\\ \frac{{\mathcal{C}}_{\epsilon }}{(\log (1/\epsilon ){)}^2}⟶\frac{2}{\pi },\end{array}$$

in probability for the discrete model and almost surely for Brownian motion (Dembo et al., 2004). The earlier post on cover times for two-dimensional random walks explains these constants and their multiscale proof.

The factor two reflects the clocks. After dividing space by $N$, the rate-one walk must be run for time $2N^{2}t$ to approximate standard Brownian motion at time $t$. This matches the leading constants. It does not by itself transfer a cover-time fluctuation theorem: covering is sensitive to the path at the shrinking target scale.

Gumbel laws and random intensities

A standard Gumbel variable $G$ has distribution

$$\mathbb{P}(G\le z)=\exp (-e^{-z}).$$

It arises, for example, from the maximum of $m$ independent mean-one exponential variables:

$$\begin{array}{c}\mathbb{P}\left(\underset{1\le i\le m}{\max }{E}_i\le \log m+z\right)\\ ={\left(1-\frac{e^{-z}}{m}\right)}^m⟶e^{-e^{-z}}.\end{array}$$

The identity holds for sufficiently large $m$ at each fixed $z$.

Now let $W$ be a positive, finite random variable. If the number of surviving targets at level $z$, conditional on $W$, is Poisson with mean $W{e}^{-z}$, then

$$\begin{array}{c}\mathbb{P}(\text{no survivor at level }z)\\ =\mathbb{E}\exp (-W{e}^{-z}).\end{array}$$

Equivalently, the last disappearance level has law $G+\log W$, with $G$ independent of $W$. A Poisson law with random intensity is called a mixed Poisson law; its spatial counterpart is a Cox process. We use this as an explanatory model, without asserting that either cover-time theorem alone establishes a full spatial Cox-process limit.

The random shift is meaningful: a larger $W$ means more persistent targets and a later cover time. A deterministic $W$ would give only a deterministic translation of the ordinary Gumbel law.

Logarithmic potential and excursions

Two-dimensional recurrence produces logarithmic hitting probabilities. In a planar annulus with radii $a<r<R$, Brownian motion started at radius $r$ satisfies

$${\mathbb{P}}_r({\tau }_a<{\tau }_R)=\frac{\log (R/r)}{\log (R/a)}.$$

Indeed, the right-hand side is the radial harmonic function with boundary values one at $a$ and zero at $R$; optional stopping identifies it with the hitting probability.

An excursion here is a segment of the path crossing between prescribed boundaries. Put concentric circles around a candidate late point, with geometrically decreasing radii. The logarithmic formula makes successive scales comparable. Counts of crossings behave like populations in a critical branching process: excursions at one scale generate excursions at the next.

If $N_j$ is such a count, the square-root coordinate $\sqrt{2N_j}$ is natural because branching fluctuations have size proportional to $\sqrt{N_j}$. This transformation approximately stabilizes their variance; the resulting scale process has Bessel-type behavior, including an absorbing state at zero. Vanishing at the final scale corresponds to an unvisited small target.

Different candidate points share much of their large-scale exploration history. Their effective branching scale is determined by their distance. This is the source of both the analogy with branching random walk and the difficulty of treating nearby late points as independent.

Gaussian free fields and critical multiplicative chaos

A Gaussian free field (GFF) is a centered Gaussian field whose covariance is a Green operator. On a torus the Laplacian annihilates constants, so one must remove that mode. For the normalization used by Zhou, the continuum field $h$ has spatial average zero and covariance

$$\mathrm{Cov}(h)=4(-\Delta {)}^{-1}$$

on mean-zero test functions. The field is a random distribution rather than a pointwise-defined function.

Smooth it at scale $\delta$ using the heat semigroup:

$$\begin{array}{c}{h}_{\delta }=e^{{\delta }^2\Delta }h,g=\frac{2}{\pi },\\ V_{\delta }:=\mathbb{E}[h_{\delta }(x{)}^2],\\ V_{\delta }=g\log (1/\delta )+O(1).\end{array}$$

Translation invariance makes $V_{\delta }$ independent of $x$. The logarithmic variance reflects contributions from many spatial scales.

To normalize the exponential of a centered Gaussian variable, one subtracts half the variance of the exponent: here the subtraction is ${\alpha }^2{V}_{\delta }/2$. At the critical parameter $\alpha =2/\sqrt{g}$, an additional square-root factor is needed. Zhou uses the random measure

$$\begin{array}{c}{\mathcal{Z}}_{\delta }(dx)=\sqrt{\log (1/\delta )}\\ \cdot \exp \left(\alpha h_{\delta }(x)-\frac{{\alpha }^2}{2}{V}_{\delta }\right)dx,\\ \mathcal{Z}=\underset{\delta \downarrow 0}{\lim }{\mathcal{Z}}_{\delta },\end{array}$$

where convergence is weak convergence of measures in probability. This is critical Gaussian multiplicative chaos. Its total mass

$$Z=\mathcal{Z}({\mathbb{T}}^2)$$

is almost surely positive and finite. It packages the random spatial variation that survives after all the small-scale fluctuations have been integrated out. The measure is not a pointwise exponential of the unsmoothed field. Its construction and normalization are treated in Zhou’s Chapter II (Zhou, 2026).

🏷️ The Two Limit Theorems

All logarithms below are natural. A limit law determines fluctuations around a center, substantially more information than a first-order asymptotic or a tightness bound.

Zhou: random walk on the discrete torus

Set $L=\log N$ and

$$\begin{array}{rl}{b}_N& =\frac{2}{\pi }{N}^{2}L,\\ a_N& =b_N(2L-\log L).\end{array}$$

Discrete torus limit law

Zhou’s Theorem VIII.7 states that, for a deterministic $\kappa >0$,

$$\frac{T_N-a_N}{b_N}⟹G+\log (\kappa Z),$$

where $G$ is standard Gumbel and independent of $Z$. Equivalently, the limiting distribution function is

$$F_{\mathrm{disc}}(z)=\mathbb{E}\exp (-\kappa Z{e}^{-z}).$$

The paper proves convergence uniformly in the starting vertex and in the distribution-function argument. The same normalization applies to the discrete-time jump chain (Zhou, 2026).

The center contains both the leading $N^2(\log N{)}^2$ term and a negative $N^2\log N\log \log N$ correction. The new theorem resolves the remaining fluctuations of order $N^2\log N$. The constant $\kappa$ depends on the normalization of $\mathcal{Z}$; only the product $\kappa Z$ enters the law.

Dembo–Rosen–Zeitouni: Brownian covering

Put $L_{\epsilon }=\log (1/\epsilon )$ and

$$m_{\epsilon }=2L_{\epsilon }-\frac{1}{2}\log L_{\epsilon }.$$

The Brownian paper rescales physical time by $2\pi$ on the unit-area torus. Define the centered variable

$$Y_{\epsilon }=\sqrt{2\pi {\mathcal{C}}_{\epsilon }}-m_{\epsilon }.$$

Keeping our physical-time notation ${\mathcal{C}}_{\epsilon }$, its theorem reads as follows.

Brownian torus limit law

Dembo–Rosen–Zeitouni’s Theorem 1.1 states that there is a random variable $0<X_{\star }<\infty$ almost surely such that

$$\begin{array}{c}\underset{\epsilon \downarrow 0}{\lim }\mathbb{P}(Y_{\epsilon }\le z)\\ =\mathbb{E}\exp (-X_{\star }{e}^{-2z}).\end{array}$$

Thus the limiting variable can be represented as

$$\frac{G+\log X_{\star }}{2},$$

with $G$ independent of $X_{\star }$ (Dembo, Rosen & Zeitouni, 2026).

Sign convention in v1

The representation following equation (1.5) in the Brownian v1 manuscript prints a minus sign before $\log X_{\star }$. Its displayed distribution function requires a plus sign: conditional on $X_{\star }$, the probability that $(G+\log X_{\star })/2\le z$ is exactly $\exp (-X_{\star }{e}^{-2z})$. We follow that distribution function throughout.

A common normalization

Let $m(L)=2L-\frac{1}{2}\log L$. Zhou’s theorem also implies

$$\begin{array}{c}2\left(\sqrt{\frac{\pi T_N}{N^2}}-m(\log N)\right)\\ ⟹G+\log (\kappa Z).\end{array}$$

The Brownian result is

$$\begin{array}{c}2\left(\sqrt{2\pi {\mathcal{C}}_{\epsilon }}-m(L_{\epsilon })\right)\\ ⟹G+\log X_{\star }.\end{array}$$

These expressions align under $T_N/(2N^2)\leftrightarrow {\mathcal{C}}_{1/N}$. To check the discrete conversion, write $T_N=a_N+b_N{Y}_N$, where $Y_N$ is tight. Then

$$\begin{array}{c}\frac{\pi T_N}{N^2}=4L^2-2L\log L+2L{Y}_N,\\ L=\log N,\end{array}$$

and taking the square root gives $m(L)+Y_N/2+o_{\mathbb{P}}(1)$. The omitted term $(\log L{)}^2/L$ tends to zero.

The normalizations therefore agree in their leading term, logarithmic correction, and fluctuation scale. This does not establish that $X_{\star }$ and $\kappa Z$ have the same law. Zhou identifies a specific critical-chaos mass; the Brownian paper constructs its mixing variable through limits of excursion-based quantities. It does not state an identification with this same chaos mass.

🏷️ The Common Extreme-Value Mechanism

At a late time, neighboring unvisited points tend to occur in clusters. Counting every point separately overcounts the number of effectively distinct obstacles to coverage. Both proofs organize the late set into spatial units and isolate two kinds of randomness:

• Large-scale exploration determines a random abundance of difficult regions.
• The remaining fine-scale exploration decides which of these regions still contains a hole.

Here is the elementary calculation underlying the common endpoint. Given a coarse history $\mathcal{F}$, suppose region $i$ retains a hole with small conditional probability $p_i(z)$. If the required conditional independence or decorrelation estimates hold, then

$$\begin{array}{c}\mathbb{P}(\text{coverage}\mid \mathcal{F})\\ \approx \prod _i(1-p_i(z))\\ \approx \exp \left(-\sum _i{p}_i(z)\right).\end{array}$$

The second approximation follows from $\log (1-p)=-p+O(p^2)$: it suffices that the largest $p_i$ tends to zero and their sum stays bounded. If their sum converges to $W{e}^{-z}$, averaging gives $\mathbb{E}{e}^{-W{e}^{-z}}$.

This calculation explains the Gumbel form. The research problem is to justify the spatial decomposition and identify $W$ with relative errors tending to zero. Bounds correct only up to fixed multiplicative constants generally cannot identify the limiting distribution.

flowchart TD
  A[Exploration across many spatial scales] --> D[Discrete: low local-time components]
  A --> B[Brownian: cities and small blocks]
  D --> E[Gaussian activity and deterministic-time recovery]
  E --> F[Critical-chaos law for the component count]
  F --> G[Survival estimates in the final time window]
  B --> C[Excursion counts, barriers, and angular data]
  C --> H[Precise conditional probabilities of uncovered blocks]
  H --> I[Gaussian limit of coarse excursion statistics]
  G --> J[Probability that no hole survives]
  I --> J
  J --> K[Randomly shifted Gumbel law]

The diagram records dependencies at the level of ideas. In particular, the discrete Gaussian comparison needs a separate recovery of deterministic observation time, while the Brownian proof needs angular information in addition to radial excursion counts.

🏷️ The Discrete Proof: From Local Times to a Random Component Count

The highlighted results below are stated in the notation of this post, with auxiliary truncations suppressed where indicated. Their proof outlines explain the main reductions; the cited sections supply the uniform estimates. For the discrete argument, begin with the uniform stationary starting distribution. Translation invariance then transfers the laws of the activity, component count, and cover time to any fixed starting vertex.

The observation time and the geometry of the late set

Zhou does not begin the final covering argument with only a bounded number of holes. Write $L=\log N$. The walk is observed at a deterministic time

$$\begin{array}{rl}{s}_N& =b_N\left(2L-\frac{3}{2}\log L\right)\\ & +\frac{9g}{32}{N}^2(\log L{)}^2,\end{array}$$

which precedes the center $a_N$. Since $N^2(\log L{)}^2=o(b_N)$, the remaining interval to the test time $a_N+z{b}_N$ is

$$\begin{array}{c}{a}_N+z{b}_N-s_N\\ =b_N\left(\frac{1}{2}\log L+z+o(1)\right).\end{array}$$

At $s_N$ there are roughly $\sqrt{L}$ relevant unvisited components. Their eventual normalized count will carry the random intensity of the limit.

The geometric selection removes points with another low-local-time point at an intermediate distance. These are components under the paper’s specified distance rule, not necessarily ordinary nearest-neighbor connected components. The following elementary lemma explains why this rule is useful.

Separation of canonical components — Zhou, Lemma III.1

Fix distance scales $0<2\rho <R$. From a vertex set $U$, remove every vertex having another vertex of $U$ at graph distance strictly between $\rho$ and $R$. Join retained vertices whose distance is at most $\rho$. Every resulting component has diameter at most $\rho$, and distinct components are separated by at least $R$.

Proof outline. Along a path of retained edges, the first vertex at distance greater than $\rho$ from the initial vertex would be at distance at most $2\rho <R$. This contradicts the removal rule. Vertices in different components cannot be within $\rho$, and the forbidden intermediate distances force separation by at least $R$.

The paper uses

$$\begin{array}{c}{\rho }_N=\frac{1}{2}{e}^{L^{1/6}},\\ R_N=N{e}^{-2L^{1/8}}.\end{array}$$

Thus local components are small compared with their mutual separation. Probabilistic estimates in Chapter IV control the discarded set; the geometric lemma alone says nothing about how many vertices are discarded.

Gaussian activity as an intermediate observable

The indicator that a component contains a zero of local time is too abrupt to transfer directly through a Gaussian identity. Zhou instead assigns a nonnegative contact activity to each low-local-time component.

Its basic meaning can be expressed without the full notation. Fix a local-time profile $\ell$, a component $D$, a Gaussian probe field $\psi$, and a small threshold $v$. Consider the scalar shifts $q$ for which

$$\begin{array}{c}2\ell (x)+(\psi (x)+q{)}^2\le v\\ \text{for at least one }x\in D.\end{array}$$

For each admissible $x$ this condition gives an interval of shifts. The expected length of their union, with a fixed normalization, is the activity. The Gaussian probe has a specific conditional covariance in the paper; it is not an arbitrary independent family of Gaussian variables.

This construction measures how readily the whole component can produce a small value in a square-field comparison. Taking a union rather than adding lengths separately respects the clustering of nearby vertices. Averaging over the probe makes activity a function of the occupation profile, not an additional source of randomness in the final theorem. Let $A_N$ be the sum of these activities at $s_N$, divided by $\sqrt{L}$. The paper takes the canonical components of $\{x:{\ell }_{s_N}(x)\le 1\}$, contact threshold $v=1/4$, and normalizes the expected interval length by $(2\pi g{)}^{-1/2}$.

The Gaussian–local-time link belongs to the family of Ray–Knight isomorphisms: distributions involving occupation fields plus squared Gaussian fields can be related to shifted squared Gaussian fields. Here the required form is an exact finite-graph coarea identity, a change-of-variables formula on field space. It initially compares integrated path statistics with Gaussian configurations constrained by their total squared norm; it is not a direct identity between a fixed-time unvisited set and GFF extremes. Chapters III–V establish the transfer needed for $A_N$ (Zhou, 2026).

Recovering deterministic time

A crucial issue is the clock. Gaussian comparison introduces a random time shift coming from a squared Gaussian coordinate. Knowing an average over such shifts does not automatically determine the law at $s_N$.

In the limiting calculation, the averaging kernel is the density of half the square of a standard normal variable:

$$k(u)=\frac{e^{-u}}{\sqrt{\pi u}}{1}_{\{u>0\}}.$$

This is a Gamma density of shape $1/2$. Its useful algebraic feature is

$$(k\ast k)(u)=e^{-u}{1}_{\{u>0\}}.$$

One-sided Gamma inversion — the mechanism in Zhou, Proposition V.5

For the kernel $k$ above, let $f$ be bounded and measurable. If $(k\ast f)(x)=0$ for every $x<a$, then $f(x)=0$ for almost every $x<a$. The convolution uses only values $f(x-u)$ with $u>0$.

Proof outline. Convolve once more. Since the kernel is supported on positive arguments, all intermediate arguments remain below $a$. The identity for $k\ast k$ and Fubini’s theorem give

$$0=e^{-x}{\int }_{-\infty }^x{e}^{y}f(y)dy,$$

for $x<a$. The integral is locally absolutely continuous; differentiating it gives $e^{x}f(x)=0$ almost everywhere. In the paper, $f$ is the difference between a subsequential Laplace-transform limit and its proposed value. The half-line available for inversion depends on the Laplace parameter.

Deterministic-time activity limit — Zhou, Theorem V.1

With the activity and observation time specified above, there is a deterministic $\Gamma >0$ such that

$$A_N⟹\Gamma Z.$$

Proof outline. The finite-graph coarea identity transfers Gaussian shell asymptotics, governed by the critical measure of Chapter II, to averaged Laplace transforms of activity. Tightness permits subsequential limits of these bounded tests. Gamma inversion identifies those limits on the available half-lines. Short-window comparisons remove the time averaging and the endpoint conditioning. Finally, uniqueness of Laplace transforms identifies the probability law at $s_N$. The Gaussian identification and the recovery of physical time are both necessary.

From activity to exact zeros

Let $K_N$ count the selected unvisited components at $s_N$. The next task is to compare their zero-local-time indicator with contact activity. Chapters VI–VII resample excursions inside local regions while preserving the exterior information, including the constraints needed to control durations. This puts both observables under common conditional local laws.

Activity determines the unvisited-component count — Zhou, Theorem VII.10

There is a deterministic $\beta >0$ such that

$$\frac{K_N}{\sqrt{L}}-\beta A_N\stackrel{\mathbb{P}}{\to }0.$$

Consequently,

$$\begin{array}{c}\frac{K_N}{\sqrt{L}}⟹\kappa Z,\\ \kappa =\beta \Gamma .\end{array}$$

Proof outline. Under common conditional local laws, compare the indicator of a zero with its Gaussian contact activity. Summing gives a comparison at any fixed accuracy, initially with a coefficient depending on local cutoffs. Placing the comparisons on a common time window shows that these coefficients have a unique limit. Increment estimates transfer the residual bound to $s_N$. Packing and local clearing show that the normalized zero count does not collapse to zero, forcing $\beta >0$. The activity limit and Slutsky’s theorem then give the stated convergence.

This identifies the random coefficient of the component count, not deterministic upper and lower bounds for $K_N/\sqrt{L}$.

The final covering window

Let $H_A$ be the first hitting time of a set $A$, and put ${\bar{u}}_N(z)=b_N(\frac{1}{2}\log L+z)$, which differs from the actual remaining interval by $o(b_N)$.

Joint survival of separated targets — Zhou, Proposition VIII.1

Fix $k$ and a bounded range of $z$. If $A$ consists of $k$ points at pairwise Euclidean distances at least $R_N/2$, and the starting point $y$ satisfies $d(y,A)\ge R_N/8$, then, uniformly over these choices,

$$\begin{array}{c}{\mathbb{P}}_y(H_A>{\bar{u}}_N(z))\\ =L^{-k/2}{e}^{-kz}(1+o(1)).\end{array}$$

Without the distance condition on $y$, the paper proves the uniform upper bound $C_{k,W}{L}^{-k/2}$ for $|z|\le W$.

Proof outline. The Green matrix on $A$ has diagonal entries of order $gL$ and much smaller off-diagonal entries. Capacity and the leading eigenvalue of the walk killed on $A$ give the survival rate $k/b_N$ to sufficient accuracy. Evaluating the exponential at ${\bar{u}}_N(z)$ produces $L^{-k/2}{e}^{-kz}$. A mixing argument transfers the stationary estimate to distant starting points while controlling early hits to relative precision; Appendix AB supplies this last estimate.

The condition on the starting point matters. In the final argument, at most one representative lies too close to the walk’s position. Remove it for the factorial-moment calculation, and use the uniform upper bound to show that its survival probability is negligible.

The scales now fit:

$$\begin{array}{c}\frac{K_N}{\sqrt{L}}⟹\kappa Z,\\ p_N(z)=L^{-1/2}{e}^{-z}.\end{array}$$

Here $p_N(z)$ is the asymptotic survival probability of a distant representative. Their product is of order one in probability and retains the random factor $K_N/\sqrt{L}$. The fixed-$k$ estimates control factorial moments, which count ordered tuples of distinct survivors. Inclusion–exclusion then identifies the probability of no survivors. The argument first restricts $K_N/\sqrt{L}$ to a bounded range and removes that restriction using tightness; it does not require finite moments of $Z$.

Hitting a representative does not yet cover its entire component. Local clearing estimates handle that difference, and separate estimates handle the discarded set. Chapter VIII obtains

$$\begin{array}{c}\mathbb{P}(T_N\le a_N+z{b}_N)\\ =\mathbb{E}\exp \left(-\frac{K_N}{\sqrt{L}}{e}^{-z}\right)+o(1).\end{array}$$

The component-count limit completes the argument. The random intensity has been identified before the last-window disappearance calculation.

🏷️ The Brownian Proof: Precise Excursion Probabilities

Cities, blocks, and an excursion clock

Dembo–Rosen–Zeitouni divide the torus into congruent coarse squares called cities, separated by narrow channels. Each city contains much smaller blocks, of side comparable to $\epsilon e^{K_5}$, where $K_5$ is a large auxiliary parameter. A block is counted if it contains an uncovered point; this groups nearby holes before applying moment estimates.

The path outside city interiors supplies a coarse record. Conditional on the appropriate exterior records, the remaining interior excursions admit the product structure used to combine city-level probabilities. The authors also replace physical time by a count of inter-city excursions.

Excursion-clock approximation — Dembo–Rosen–Zeitouni, Lemma 2.5

Fix the city size and channel parameter. Let $S_m$ be the time required for $m$ inter-city excursions, and let $\mu$ be their stationary mean duration. As $m\to \infty$,

$$\sqrt{S_m}-\sqrt{m\mu }⟹\mathcal{N}(0,{\sigma }^2).$$

The limiting variance tends to zero as the city size tends to zero, with the channel parameter fixed.

Proof outline. Positivity of the boundary transition densities gives a uniformly mixing Markov chain for excursion endpoints and durations. Apply its central limit theorem to $S_m$ and then linearize the square root. Folding the torus onto a single congruent tiling square and using Brownian scaling shows that the resulting variance vanishes as the tiling is refined. The order of limits is essential: at a fixed tiling, the clock error need not vanish in the limiting square-root coordinate.

This is a precision statement, not merely a law of large numbers. An error small relative to ${\mathcal{C}}_{\epsilon }$ could still be large compared with the order-one fluctuations of its normalized square root. The congruent tiling of the torus is essential to the clock and coarse-field arguments in Sections 2.2 and 8 (Dembo, Rosen & Zeitouni, 2026).

Barriers and the modified second moment method

Around each block, count excursions through concentric circles. A block that remains uncovered has an unusually small fine-scale count. But counting all such candidates directly gives a poor second moment, because nearby blocks can owe their survival to the same exceptional large-scale history.

The proof retains candidates whose square-root excursion counts stay above a carefully chosen barrier at intermediate scales, while still allowing an uncovered point at the end. Such barriers suppress histories that create excessive correlations. The authors prove separately that this truncation loses asymptotically negligible covering probability.

This is the modified second moment method: first choose a restricted family of rare events whose pair correlations are tractable, then show that the restriction captures the original extreme event. Pair estimates depend on the scale at which two blocks separate. Sections 4 and 7 treat the barrier reduction, second moments, and tightness.

There is an important geometric difference from a tree. Two nearby centers in the plane do not have exactly the same excursion counts at their shared coarse scales. Continuity and coupling estimates must replace the exact common ancestry available in branching models.

Why angular information survives conditioning

The annulus formula concerns radial hitting probabilities. Once the exterior path has been fixed, however, the entry and exit locations of an interior excursion are also constrained. A Brownian bridge with prescribed endpoints does not have the same radial law as an unconditioned excursion.

The proof therefore tracks an empirical distribution of angular increments, together with excursion counts. Its deviation from equilibrium gives an angular defect. This information is needed both for decoupling nearby blocks and for computing their conditional survival probabilities at the required precision.

At the level of the main asymptotic, the relevant weight has the form

$$w(y,v)=y{e}^{-2y}{e}^{-2v},$$

where $y$ is the centered square-root excursion coordinate and $v$ is a scaled angular correction. To state the local result compactly, let $s_{\mathrm{b}}=\epsilon e^{K_5}$ be the block side length, $r_{\mathrm{c}}$ the coarse excursion radius, and ${\mathcal{U}}_{\mathrm{b}}$ the paper’s retained block-uncovering event, including its barriers and angular restrictions.

Precise conditional block probability — Corollary 2.17, compressed notation

Let $\mathcal{E}$ denote the endpoint data. On the prescribed admissible windows,

$$\begin{array}{rl}& \mathbb{P}({\mathcal{U}}_{\mathrm{b}}\mid \mathcal{E})\\ & =(1+o(1)){\left(\frac{s_{\mathrm{b}}}{r_{\mathrm{c}}}\right)}^2\\ & \cdot c_{\mathrm{loc}}w(y,v).\end{array}$$

Here $c_{\mathrm{loc}}>0$ is deterministic and depends on the local cutoffs. The error tends to zero uniformly in the allowed block and endpoint data, in the paper’s prescribed order of limits. This is not a formula for arbitrary, untruncated excursions.

Proof outline. Proposition 2.14 separates the angular conditioning at the coarse scale from the terminal uncovering event and the deeper excursion counts. Proposition 2.16 evaluates the remaining barrier expectation using conditional local central limit estimates. Combining them produces the geometric area ratio, the local coefficient, and the weight $y{e}^{-2y}{e}^{-2v}$. The decisive improvement over rough tail bounds is the relative error $1+o(1)$: a constant-factor error would leave the final intensity undetermined.

Conditional first moments determine the limit

Let ${\Lambda }_u(z)$ count retained uncovered blocks in city $u$, and let $\mathcal{F}$ denote the conditioning data used for the city decomposition. Define

$$\begin{array}{rl}{q}_u(z)& =\mathbb{E}[{\Lambda }_u(z)\mid \mathcal{F}],\\ r_u(z)& =\mathbb{E}[({\Lambda }_u(z){)}_2\mid \mathcal{F}].\end{array}$$

Here $(n{)}_2=n(n-1)$. These are conditional first and second factorial moments. The elementary bounds

$$\begin{array}{c}{q}_u(z)-\frac{1}{2}{r}_u(z)\\ \le \mathbb{P}({\Lambda }_u(z)\ge 1\mid \mathcal{F})\\ \le q_u(z)\end{array}$$

show why controlling pairs allows the first moment to approximate the probability that the city contains a hole.

The precise block calculation identifies the $q_u$. Its coarse random inputs still need a limit law. Denote the normalized city weights, after the paper’s truncations, by $X_{u,z}^{(\epsilon )}$. Their aggregate approximates the conditional first moments after the factor $e^{-2z}$ is extracted.

Limit of the coarse city weights — Proposition 2.24

At fixed admissible auxiliary parameters, as $\epsilon \downarrow 0$,

$$\begin{array}{c}(X_{u,z}^{(\epsilon )}{)}_{u=1}^M\\ ⟹(X_u{)}_{u=1}^M,\end{array}$$

where $M$ is the number of cities and the limiting joint law is independent of the fixed test parameter $z$.

Proof outline. Center and normalize the excursion counts and angular defects accumulated over the inter-city record. Markov-chain central limit estimates identify their finite-dimensional Gaussian limits. Continuity estimates in the spatial center allow these limits to be used in averages over the many small blocks. Applying the truncated weight function and the relevant conditional expectations gives the city-weight limits. The Gaussian objects are the underlying coarse fields; the nonlinear weights $X_u$ need not themselves be Gaussian.

Conditional moments identify the no-hole probability — Theorem 2.12 and Proposition 2.25

The summed conditional second factorial moments satisfy, with probability tending to one in the ordered limits,

$$\begin{array}{c}\sum _u{r}_u(z)\\ \le \xi \left(1+\sum _u{q}_u(z)\right),\\ \xi ⟶0.\end{array}$$

Together with the precise first-moment asymptotic and vanishing individual city probabilities, this makes the expected product of city no-hole probabilities agree asymptotically with

$$\mathbb{E}\exp \left(-\sum _u{q}_u(z)\right).$$

The latter is identified through the Laplace transforms of $\overline{X}{e}^{-2z}$, where $\overline{X}={\sum }_u{X}_u$ still depends on the auxiliary parameters.

Proof outline. Classify pairs of blocks by their separation scale. Barrier and decoupling estimates make their total contribution small relative to one plus the first moment; Section 7 treats early, intermediate, and terminal separation separately. The elementary factorial-moment bounds above then replace each city’s hole probability by $q_u$. Since individual probabilities vanish, the product becomes an exponential of the summed first moments. Proposition 2.24 and the precise block asymptotic identify that sum through the city weights. The cutoff dependence is removed only in the final tightness and limit argument.

Combining cities then gives, schematically,

$$\begin{array}{c}\mathbb{P}(\text{coverage at level }z)\\ \approx \mathbb{E}\prod _u(1-q_u(z){)}_{+}\\ \approx \mathbb{E}\exp \left(-\sum _u{q}_u(z)\right)\\ ⟶\mathbb{E}\exp (-X_{\star }{e}^{-2z}).\end{array}$$

Here $(a{)}_{+}=\max (a,0)$, since a conditional first moment need not itself be a probability. The construction of $X_{\star }$ uses normalized limits of sums of city weights, described by the authors as being of derivative-martingale type. Tightness of the centered cover times (Proposition 2.26) excludes mass at zero or infinity in the limiting intensity, while the Laplace-transform identification makes its law unique. Calling these weights derivative-martingale type does not identify them with a specified GFF chaos measure.

The limits have a deliberate order: first $\epsilon \downarrow 0$, then the coarse and fine auxiliary scales are sent to their limits, and finally the channel-width parameter is removed. Boundary zones, truncation errors, and the excursion-clock error are controlled along this passage. Simply deleting channels because they have small area would be insufficient: the last uncovered point could lie in a small exceptional region.

🏷️ Comparison of the Technical Contributions

Feature	Zhou: discrete random walk	Dembo–Rosen–Zeitouni: Brownian motion
Target	Every vertex of $V_N$	Every radius-$\epsilon$ neighborhood on ${\mathbb{T}}^2$
Rare spatial unit	Geometrically selected low-local-time or unvisited component	Small block containing an uncovered point
Main intermediate statistic	Gaussian contact activity	Conditional expected number of retained uncovered blocks
Gaussian input	GFF comparison and critical multiplicative chaos	Gaussian limits of excursion counts and angular defects
Clock issue	Invert Gaussian-induced time averaging and recover deterministic time	Compare inter-city excursion counts with physical time
Main local comparison	Activity versus the presence of an exact zero of local time	Precise barrier-conditioned block probabilities with angular data
Final calculation	Joint survival of representatives, followed by local clearing	City-level product, with conditional factorial-moment control
Description of the shift	$\log (\kappa Z)$ for a specified critical-chaos mass	$\log X_{\star }$ from a limit of normalized excursion-based weights

The shared structure is substantial: logarithmic correlations, multiscale barriers, clustering, a random coarse intensity, and a final no-survivor probability. These features explain why the two results belong to the same extreme-value picture.

The technical distinction is equally substantial. Zhou’s principal chain is Gaussian activity → deterministic-time law → component count → final coverage. The Brownian chain is conditioned excursions → precise block probabilities → Gaussian coarse-field limit → final coverage. The papers do not supply a routine discrete-to-continuum reduction between these chains, nor do their stated theorems establish equality of the two mixing distributions.

The common lesson is that identifying the last holes requires more than estimating their typical number. One needs the distribution of their coarse-scale abundance and enough local precision to turn that abundance into an exact limiting no-hole probability.

🏷️ Reading Guide to the Papers

For Zhou, the introduction and Chapter VIII show the overall reduction most directly. Chapters III and V explain the activity observable and deterministic-time recovery; Chapters VI–VII explain why activity measures the number of genuinely unvisited components. Chapter II supplies the critical-chaos input. The appendices contain the detailed uniform estimates supporting these interfaces.

For Dembo–Rosen–Zeitouni, Sections 1.2 and 2 provide the proof architecture. Sections 5–6 contain the precise conditional first-moment calculation; Section 7 supplies second moments and tightness; Section 8 constructs the limiting coarse field. Sections 3, 4, and 9 supply the barriers, couplings, continuity, and approximation estimates needed to justify those reductions.

See Also

• on cover times for two-dimensional random walks: The first-order constants, logarithmic hitting probabilities, and the original successful-point construction provide the starting point for these fluctuation theorems.
• on high-dimensional cover times and random interlacements: Higher-dimensional tori exhibit ordinary Gumbel fluctuations, providing a useful comparison with the random intensity retained in dimension two.
• on Slepian’s lemma and Gaussian comparison: Background on comparing correlated Gaussian fields; such comparisons motivate the GFF connection, while the local-time and clock arguments here require additional structure.
• on Dudley’s Theorem: A complementary account of multiscale Gaussian analysis, useful for understanding why controlling a field across scales is different from estimating it at one point.

References

🐻  Dembo, A., Peres, Y., Rosen, J. & Zeitouni, O. 2004. Cover Times for Brownian Motion and Random Walks in Two Dimensions. Annals of Mathematics 160(2), 433–464.

🐻  Dembo, A., Rosen, J. & Zeitouni, O. 2026. Limit Law for Brownian Cover Time of the Two Dimensional Torus.

🐻  Zhou, Y. 2026. A Limit Law for the Cover Time of the Two-Dimensional Discrete Torus.