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arxiv: 2604.14122 · v1 · submitted 2026-04-15 · 🧮 math.PR · math-ph· math.MP

The scaling limit of random walk and the intrinsic metric on planar critical percolation

Irina {\DJ}ankovi\'c , Maarten Markering , Jason Miller , Yizheng Yuan This is my paper

Pith reviewed 2026-05-10 12:20 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords critical percolationscaling limitsCLE6SLE6random walkchemical distanceEinstein relationsconformal loop ensemble
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The pith

The simple random walk on critical percolation clusters converges in the scaling limit to Brownian motion supported on the CLE6 gasket.

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

The paper proves that simple random walk on the open clusters of critical site percolation on the triangular lattice, when suitably rescaled, converges to a continuous diffusion process that remains inside the gasket of the conformal loop ensemble with parameter 6. It further establishes that the intrinsic chemical distance on these clusters converges to the geodesic metric defined by the same CLE6 structure. These two limits together imply the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension for the clusters. The exponents are shown to satisfy the Einstein relations that link diffusion, resistance, and dimension.

Core claim

The simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter κ = 6 (CLE6), the so-called CLE6 Brownian motion. The intrinsic (chemical distance) metric on these clusters converges in the scaling limit to the geodesic CLE6 metric. As a direct consequence the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters exist and satisfy the Einstein relations.

What carries the argument

The gasket of the conformal loop ensemble with κ = 6 (CLE6), which supports both the limiting CLE6 Brownian motion for the rescaled random walk and the limiting geodesic metric for the intrinsic distance.

If this is right

  • The chemical distance exponent for critical percolation clusters exists.
  • The resistance exponent and spectral dimension of the clusters also exist.
  • These three exponents obey the Einstein relations connecting diffusion, resistance, and dimension.

Where Pith is reading between the lines

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

  • Large-scale geometry and transport on percolation clusters can now be studied directly through the continuous CLE6 gasket rather than through lattice approximations.
  • The same convergence technique may extend to other planar critical models whose interfaces are described by SLE6.
  • Numerical sampling of CLE6 can be used to predict discrete percolation exponents without simulating the lattice model itself.

Load-bearing premise

The interfaces of critical percolation converge to SLE6 curves, so that the associated CLE6 gasket and its geodesic metric are well-defined and regular enough for the random-walk and metric convergences to hold.

What would settle it

A numerical check on large triangular-lattice percolation configurations showing that the rescaled random-walk paths fail to converge in distribution to the expected diffusion on the CLE6 gasket, or that the rescaled graph distances fail to approach the geodesic CLE6 metric.

Figures

Figures reproduced from arXiv: 2604.14122 by Irina {\DJ}ankovi\'c, Jason Miller, Maarten Markering, Yizheng Yuan.

Figure 3.1
Figure 3.1. Figure 3.1: An illustration of the event E. The blue curve represents an open crossing from left to right. The red curves represent two closed left-to-right crossings that come within distance 2 but do not intersect. if they exist. Note that the items (iii)–(v) in the definition of the event E(Λn) imply that the distance D(u, v) does not depend on the percolation configuration within the 2δ-neighborhoods of c (0) 1 … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Four points zi1, . . . , zi4 together with an annulus such that the pairwise path diameter distances are at least the size of the annulus. Because d path n (zi1, zi2) is greater than the diameter of the annulus, it must be that any open path connecting these two points leaves the annulus. Moreover, since these points are in the same cluster, such an open path exists. In particular, we can find 2 disjoint… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: An illustration of events E1(Λn) and E2(Λn, S). The red and blue interface on the bottom of the second picture belong to the set S - this is the boundary of the closed cluster adjacent to the bottom. Note that for any possible S, P(E2(Λn, S)) ≥ P(E2(Λn, ∅)), and the latter is bounded away from zero by a constant c2 = c2(δ) > 0, again by standard RSW estimates. Moreover, the event E1(Λn) ∩ {S = S} ∩ E2(Λn… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: The figure above illustrates the event Ac4(b, B) for a (not necessarily centered) annulus B \ b. Green segments on the outer (resp. inner) boundaries represent the segments of length 2δR (resp. 2δr) centered at the midpoints of the 4 sides of this boundary. Red and blue curves represent the closed (resp. open) arms crossing this annulus. Note that for all 1 ≤ j ≤ 4, ηj starts at ij and ends at Ij . Moreo… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: An illustration showing the event that a box B of level 3 is important. The blue and red curves represent the open (resp. closed) crossings of our annuli. Note that these curves come close to centers of all superboxes of B, except for B(1) . The two dashed red curves represent the closed horizontal crossings in the outermost annulus. The two dashed red lines represent the closed horizontal crossings in t… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: An illustration of the event that a box B of level 3 is B(1)-bad. If we wanted B to be B(0)-bad, the crossings in the final annulus B(1) \ B(2) should be replaced by crossings in B(0) \ B(2), which now need not come close to midpoints of the sides of B(1), but, they must come close to midpoints of the sides of B(0). If in addition we wanted B to be bad, we would need to include two red left-to-right cros… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: An illustration of events H and Ac4  B (2) δ , B(1) δ  near the point c (1) 1 . The red curve represents the closed arm η1 with an extension γ coming from the event Ac4  B (2) δ , B(1) δ  . The green curve is the closed circuit inside of the annulus A(c (1) 1 ; δn−1Rκ−1 , 2δn−1Rκ−1 ), whose existence is guaranteed by H. H also gives us the existence of the three purple curves, which represent the ver… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: On the left-hand side are pictured two closed loops with the least amount of defects – the dotted and the dashed red lines. The full red line represents the concatenation of the outermost parts of these loops. On the right-hand side picture, the red loop represents the outermost loop ϕ. Note that this loop can be self-intersecting. The blue sites represent the defects v1, . . . , vm. One can also see the… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: A sketch of the events F c 2 in the top-right corner, F31 on the left and F32 on the bottom. On F c 2 , there are 2 closed arms with defects (blue points on the picture above) and 1 open arm from the sides of the box going all the way to the boundary. These arms are contained in a wedge domain of angle at most 5π/3 as pictured above. On F31, there are 6 open blue arms from the sides of the box to the bou… view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Figure on the left illustrates the event F c 4 in the case that ϕi does not intersect the boundary of B(x, 2r). Yellow boxes above represent the closed boxes. Blue curves represent the open crossings ηi and ηi+1 and the area containing ϕi that is enclosed by these crossings is the relevant sector-like domain. The second figure depicts the events F40 on the bottom, F41 on the top and F42 on the right. Sim… view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: The first of the three figures illustrates the event F5(b) c - if clusters of ϕi−1 and ϕi do not touch outside of the box B(vi , br), then the blue arms pictured above must be disjoint. The second figure pictures the event F6(c) c in the case that continuation of the path η revisits the box B(vi , c1/3 r). Last figure illustrates the same event, but in the case that η does not come back inside B(vi , c1/… view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: The blue-red curves on the picture represent the explored cluster boundaries starting from ∂B(w, 2r˜). The two curves reaching the inner boundary are γ and γ ′ . The yellow curves represent the closed circuits in A [PITH_FULL_IMAGE:figures/full_fig_p054_5_10.png] view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: The figure above illustrates how one constructs the 4 thinner corridors in the case that neither ϕi nor ϕj intersect ∂B(x, 4n −1Rκ−f ). The red/blue interfaces represent the curves ϕi , ψi and ϕj , ψj , while the dashed line represents the remaining portion of ϕ. The yellow boxes illustrate the modified corridors C ′ i and C ′ j that join at some point (in the case above Bi 6 = B j 4 ). Note that the fi… view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: This figure illustrates how we find the 2 corridors corresponding to ϕi and ψi in the case that ϕi intersects ∂B(x, 4n −1Rκ−f ). The two yellow boxes represent the box Bi 1 that is the single box of corridor C ′ i and 2 3Bi 1 that is the single box in the original corridor Ci (the picture is not to scale). The green corridor is the one containing the endpoint of ψi belonging to ∂B(x, 4n −1Rκ−f ). On the… view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: This figure illustrates the 4 corridors connecting different segments of ϕ to ∂B+ - these are represented by the dotted boxes. The many thinner blue and red curves represent the sequences of open and closed clusters in the annuli around these boxes that we will use in our resampling procedure. In particular, orange points are the closed sites where two consecutive open clusters touch. We will resample o… view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: The figure above illustrates the construction of the string of open clusters corresponding to ψi in the case that ψi ends at a point z on ∂B(x, 4n −1Rκ−f ). The red curve above represents ϕi , while the full blue curve represents ψi . The dashed blue curve represents the open cluster boundaries adjacent to ϕi that are not a part of ψi . The figure above also shows the box B1, which is the first box in t… view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: Left picture above illustrates the 6 arm event from the proof of part (a) of the lemma. On this picture, we have chosen to illustrate the events that z1 and z2 are touch points of macroscopic clusters using the 4 alternating arms emanating from both these points – no matter if z⋆ is a touch-point of closed or open clusters, these clusters along with their boundaries form 4 arms of alternating color – th… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: An illustration of the linking construction from Lemma 6.1. The blue line represents the open path covered by good boxes. Black points in each of the annuli represent the sets V (A1), V (A2) and V (A3) of our ‘defects’ and the red loops connecting them represent the associated ‘almost closed loop’ in each annulus. Step 2b: The bad box cannot have 5 arms. Now assume that both interfaces enter the bad box.… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Two examples of both interfaces entering the bad box. In both cases, the box has 5 arms going far. box is used, it can contribute at most (M1 + 1) · (M1+1)−2 4 qn to the distance. Thus, D(xi , xi+1) ≤ 1 2 qn + 2M1(M1 + 1) · (M1 + 1)−2 4 qn ≤ qn, where the first term comes from the good boxes and the last term comes from the bad box as explained above. It follows that P  ∃xi , xi+1 : Xxi,xi+1 ≥ qn, R−(d+… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: On the left-hand side, the area shaded in yellow represents the set Γ1. The picture on the right illustrates why the set Γ1 is not local. The area shaded in yellow once again represents the true set Γ1. The area shaded in green consists of bubbles bounded in between the two closed clusters represented by the red curves that neighbor the green region. These closed clusters are fully surrounded by the open… view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: On the picture above z is a center of an R-adic box of level i + 3 in B(k) . One can also see a box B of level j ≥ i + 9 inside B(i) = B(z, R−i/2). The pink and yellow annuli around the box B represent the two annuli in which we have 6-arm events. Then B has 6 semiarms of length j − i − 4. Lemma 6.3. The following holds for all R large enough. For any ν as in the definition of events Hj , there exists C … view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: The shaded green area on the image represents the explored part of the open cluster; orange curve represents the closed boundary of the explored region. The red curves represent the closed boundaries of the bubbles created by closed clusters adjacent to the top and bottom of B(k) , while the blue curve represents an open left-to-right path delimiting these clusters in the unexplored domain. This figure i… view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: As before, the orange and green curves represent the closed boundary of the explored region and an open path inside that region respectively. The red curves are again the unexplored boundaries of closed clusters creating bubbles. This figure depicts the target bubble with endpoints a and b. The black dashed line represents a sequence γ of sites ‘equidistant’ to the boundaries of the bubble γt and γb that… view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: This figure illustrates the case when there is a Whitney box B of level j covering the path σt that is near the boundary of the ambient box B(k) . In the case above, the top boundary of the bubble γt intersects R3B. We can see that the pink semiannulus above has three arms crossing it - two closed ones corresponding to γt and one open one coming from σ t . The yellow semiannulus is the smallest one whose… view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: The picture above portrays the annulus Aˇ with all of its crossing clusters. In this case there are 6 crossing clusters - 3 open and 3 closed ones. The red/blue curves represent the interfaces between these clusters. The endpoints of these interfaces on the inner boundary of Aˇ are denoted by xi-s and yi-s. The purple and yellow shaded boxes represent the domain Dcor. Yellow boxes represent the loop part… view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: On the figure above, we can see a sector of the annulus Aˇ together with 4 interfaces between consecutive closed cluster represented by the red/blue curves. These interfaces delimit the clusters Ci , Oi and Ci+1, as shown. The region shaded in gray represents Dcor. In this instance, we have 3 different levels within the loop part of Dcor. Green, purple and yellow shaded regions represent three corridors … view at source ↗
Figure 6.10
Figure 6.10. Figure 6.10: The figure above illustrates the case when a box B that is nice in B(k) intersects B  c (k) 1 , 2δn−1Rκ−k  . In this case, there are 8 crossing clusters in total, and C4, O4, C1 are the ones that intersect B  c (k) 1 , 2δn−1Rκ−k  . Thus m0 = 3. The area shaded in gray again represents the domain Dcor. The two smaller purple corridors represent the corridors that are constructed in the first stage of… view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: This figure illustrates an instance of the event (J 1 k ) c . The blue curve represents an open path of diameter at least n −1Rκ−k . This path must leave the box B [PITH_FULL_IMAGE:figures/full_fig_p100_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: This figure depicts the first step of our modification of the path γ connecting x and y. The original path γ is represented by the purple curve. This path is contained inside the reference box Bref marked above. Points xk1 and yk1 are the first exit points of γ from the strip S  n −1R κ−(1+ν)( 2 3 −2ν)k1+3 (shaded in yellow above) starting from x and y respectively. One can see that the segment [xk1 , … view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: This figure illustrates the compatibility property for regions V ⊆ V ′ in C. The black loops in the picture above represent the loops of Γ. The set V is represented by the green area; its boundary consists of segments of finitely many loops G ⊆ Γ. The set V ′ ⊇ V consists of the blue and green areas above; we can see that any point u ∈ V ′ \ V is separated from x and y in V ′ by exactly one of the red po… view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: The figure above illustrates the construction of the translation-invariant set Ue∗ that equals U ∗ with positive probability. The figure portrays a set U whose boundary is represented by the bold black curve above. The set U ′ ∈ Q is such that U ⊆ U ′ , and A′ is the set of all points that are not in U ′ but are within ‘triangular’ L 1 -distance ε from U ′ . The outermost loop L surrounding the origin is… view at source ↗
read the original abstract

We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $\kappa = 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves that, for critical site percolation on the triangular lattice, the simple random walk on open clusters converges in the scaling limit to a continuous diffusion (CLE6 Brownian motion) supported on the gasket of the conformal loop ensemble with κ=6. It further establishes convergence of the intrinsic chemical-distance metric to the geodesic metric induced by the same CLE6, and deduces the existence of the chemical-distance exponent, resistance exponent, and spectral dimension, together with verification of the Einstein relations among them.

Significance. If the central convergence statements hold, the work supplies the first rigorous scaling-limit description of both the random walk and the intrinsic metric on critical percolation clusters, directly linking them to the established SLE6/CLE6 framework. This yields the existence of several long-conjectured exponents and confirms the expected scaling relations, constituting a substantial advance in the geometric understanding of two-dimensional percolation.

major comments (2)
  1. [Introduction and Section 2 (CLE6 construction and gasket properties)] The central claims require that the CLE6 gasket admits a well-defined diffusion (via a resistance form or Dirichlet form) and that the geodesic metric is well-defined and continuous. The manuscript invokes these objects directly from the SLE6 construction; it is not clear from the argument whether the necessary regularity properties (local connectedness, positive capacity, Hölder continuity of paths, uniform resistance bounds) are fully established in the cited prior literature or are proved here under the assumptions available from Smirnov’s theorem. This point is load-bearing for both the random-walk and metric convergence theorems.
  2. [Section 5 (consequences and exponents)] The deduction of the chemical-distance, resistance, and spectral-dimension exponents (and the verification of the Einstein relations) is presented as an immediate consequence of the scaling limits. The manuscript should supply explicit error-control statements showing that the exponents are indeed obtained without additional assumptions on the modulus of continuity of the limiting objects.
minor comments (2)
  1. [Throughout] Notation for the gasket and the intrinsic metric should be introduced once and used consistently; the current alternation between “CLE6 gasket” and “percolation gasket” is occasionally confusing.
  2. [Section 5] A short table summarizing the exponents obtained and the Einstein relations they satisfy would improve readability of the consequences section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The two major comments concern the foundational regularity of the CLE6 objects and the rigor of the exponent derivations. We address each point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Introduction and Section 2 (CLE6 construction and gasket properties)] The central claims require that the CLE6 gasket admits a well-defined diffusion (via a resistance form or Dirichlet form) and that the geodesic metric is well-defined and continuous. The manuscript invokes these objects directly from the SLE6 construction; it is not clear from the argument whether the necessary regularity properties (local connectedness, positive capacity, Hölder continuity of paths, uniform resistance bounds) are fully established in the cited prior literature or are proved here under the assumptions available from Smirnov’s theorem. This point is load-bearing for both the random-walk and metric convergence theorems.

    Authors: The required regularity properties of the CLE6 gasket (local connectedness, positive capacity, Hölder continuity of paths, and uniform resistance bounds) are established in the prior literature on SLE6/CLE6, specifically in the works of Sheffield and Werner on the gasket and in the resistance-form constructions of Barlow, Bass, and others. Our Section 2 cites these results and invokes them under the convergence to CLE6 guaranteed by Smirnov’s theorem. To make the dependence explicit, the revised manuscript expands the discussion in the introduction and Section 2 with direct references to the precise theorems supplying each property, without reproving them here. revision: yes

  2. Referee: [Section 5 (consequences and exponents)] The deduction of the chemical-distance, resistance, and spectral-dimension exponents (and the verification of the Einstein relations) is presented as an immediate consequence of the scaling limits. The manuscript should supply explicit error-control statements showing that the exponents are indeed obtained without additional assumptions on the modulus of continuity of the limiting objects.

    Authors: We agree that the passage from scaling limits to exponents benefits from explicit error controls. The revised Section 5 will include a new paragraph that records the modulus-of-continuity estimates for the limiting diffusion and geodesic metric, obtained from the tightness arguments in Sections 3 and 4. These controls justify the extraction of the exponents and the verification of the Einstein relations directly from the convergence statements. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior CLE6/SLE6 constructions; new RW and metric limits derived independently

full rationale

The derivation assumes the known scaling limit of percolation interfaces to SLE6 (Smirnov) and the existence/regularity of the CLE6 gasket and geodesic metric as background. The central results—convergence of simple random walk on open clusters to CLE6 Brownian motion and convergence of the intrinsic chemical metric to the geodesic CLE6 metric—are stated as new theorems whose proofs are not shown to reduce by construction to these inputs. No equations equate a fitted quantity to a renamed prediction, no self-definitional loop appears in the scaling-limit statements, and the Einstein relations are deduced as consequences rather than presupposed. Self-citations to earlier CLE work by overlapping authors exist but are not load-bearing for the new convergences; the paper treats CLE6 objects as externally defined. This yields a low circularity score with no quoted reduction of the target claims to the assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters or new invented entities. It rests on the established existence and regularity properties of CLE6 and its gasket metric, which are treated as domain assumptions imported from prior work.

axioms (1)
  • domain assumption The conformal loop ensemble CLE6 exists and possesses a well-defined gasket together with a geodesic metric that supports the limiting diffusion and distance statements.
    Invoked to define the target objects of the scaling limits for both the random walk and the intrinsic metric.

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