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Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that renormalized approximations to the chemical-distance and resistance metrics on non-simple CLE gaskets are tight, and that every subsequential limit is a genuine CLE metric satisfying the same axioms.

desk verdict Serious paper with a genuine but fixable gap: the good-scheme normalization in Definition 1.6 is never proved comparable to the median used in the proofs, and the resistance-metric verification is deferred to a companion paper. read the letter →

arxiv 2507.15589 v1 pith:FXKIILFB submitted 2025-07-21 math.PR

Valeria Ambrosio , Jason Miller , Yizheng Yuan This is my paper · ORCID
classification math.PR MSC 60J6760D0560B10
keywords conformalloopensembleCLEgasketchemicaldistancemetriceffectiveresistancetightnessGromov-HausdorfftopologySchramm-Loewnerevolutionimaginarygeometry
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 studies how to put a metric on the random fractal left behind by a conformal loop ensemble when the loops are allowed to intersect themselves and each other ($\kappa'\in(4,8)$). Its central claim is that every approximation scheme for the chemical-distance or effective-resistance metric that satisfies a short list of axioms is tight after renormalization, and that every subsequential limit is again a metric of the same type, now evaluated at scale zero. The axioms include separability, Markovianity, translation invariance, compatibility, monotonicity, a series law, and a generalized parallel law; the normalization is the median of the distance across a region bounded by two intersecting loops. The proof shows that the probability of a crossing at Euclidean scale $\delta$ exceeding that median decays faster than any power of $\delta$, by bootstrap between an intersection-crossing exponent and a bubble-crossing exponent. If correct, this gives the first continuum construction of intrinsic and resistance metrics on non-simple CLE gaskets, the objects conjectured to be the scaling limits of chemical distance and resistance for models such as critical percolation.

What carries the argument

The central object is an approximate CLE$_{\kappa'}$ metric: a family of random internal metrics $d_\epsilon^V$ on gasket regions $V$, coupled to the loop ensemble, obeying a series law (distance through a separating point is at least the sum of the two sub-distances), a generalized parallel law with constants $c_s\ge 0$ and $c_p(N)>0$, plus compatibility and monotonicity conditions that control how internal metrics change when the ambient region grows. The normalization $m_\epsilon$ is the median of the metric distance across a region bounded between two intersecting CLE loops. The proof's engine is a bootstrap showing that the probability that this crossing distance at Euclidean scale $\delta$ exceeds $m_\epsilon$ decays superpolynomially in $\delta$; starting from an a priori polynomial bound with exponent $d_{\mathrm{dbl}}$ (the double point dimension of $\mathrm{SLE}_{\kappa'}$), the argument alternates between an intersection-crossing exponent and a bubble-crossing exponent, using SLE/GFF flow-line exploration, resampling of the CLE, and the series/parallel laws to gain a power improvement at each step. The Gromov-Hausdorff-function topology, defined in Appendix A, views the gasket as an abstract space with prime ends so that double points of loops correspond to distinct points, and records which finite sets separate points.

What would settle it

Walk the proof's main estimate: for a concrete approximation scheme (e.g., effective resistance on a graph approximation of the CLE$_6$ gasket), compute the median $m_\epsilon$ of the distance across a region between two intersecting loops and test whether the probability that the distance across a bubble of diameter $\delta$ exceeds $m_\epsilon$ decays faster than every power of $\delta$. If the decay is only polynomial, or if two subsequential limits can be produced that are not a.s. equal, then Theorems 1.12–1.14 would fail for that scheme; a concrete way to try is to check whether the compatibility axiom holds with $c_s>0$ for the resistance approximation, since the paper's claims are conditional on it.

Watch

Extended reading notes

Core claim

The paper proves Theorem 1.12: for any good approximation scheme (Definition 1.6), the renormalized internal metrics $m_{\epsilon}^{-1} d_\epsilon^V(\cdot,\cdot;\Gamma)$ are tight in the Gromov-Hausdorff-function topology, uniformly over a countable collection of regions $V$. Theorem 1.13 then says that the metric constructed from any subsequential limit in Section 6 is a CLE$_{\kappa'}$ metric in the sense of Definition 1.5 with $\epsilon=0$, meaning it satisfies the same axioms without approximation error. Theorem 1.14 states that any such limit is either almost surely identically zero or almost surely a true metric with $d^V(x,y)>0$ for $x\neq y$. For geodesic approximation schemes, Theorem 1.17 upgrades the limit to a non-degenerate geodesic metric whose distance is the infimum of the limiting length over admissible paths. The proofs also yield H\"older continuity of the limiting metrics with respect to the path metric, and on the thin gasket with respect to the Euclidean metric.

Load-bearing premise

The whole proof depends on the axiom system of Definition 1.5 holding at every scale down to the approximation size, especially the series law and the generalized parallel law; the paper defers the verification of these axioms for resistance approximations to a separate paper.

Editorial extensions

If this is right

  • If a family satisfies the axioms of Definition 1.5 and the good-scheme conditions (1.4)–(1.5), then for every sequence $\epsilon_n\to 0$ there is a subsequence along which the renormalized internal metrics $m_{\epsilon_n}^{-1}d_{\epsilon_n}^V$ converge in law.
  • Every subsequential limit is a CLE$_{\kappa'}$ metric with $\epsilon=0$; in particular the internal metrics are determined by a countable collection, are compatible under region restriction, and satisfy the Markovian property.
  • A CLE$_{\kappa'}$ metric is degenerate if and only if it is identically zero on every region; otherwise it is a genuine metric with positive off-diagonal distances.
  • For geodesic schemes, the limiting metric is geodesic and non-degenerate: $d^V(x,y)=\inf_{\gamma\in\mathcal{P}(x,y;V;\Gamma)} L_{d^D}(\gamma)$.
  • The limiting metrics are H\"older continuous with respect to the path metric, and on the thin gasket with respect to the Euclidean metric; the associated scaling constants satisfy $r^{d_{\mathrm{SLE}}+o(1)}m_\lambda \le m_{r\lambda}\le r^{d_{\mathrm{dbl}}+o(1)}m_\lambda$.

Reading between the lines

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

  • If the deferred verification in [MY25a] goes through, the resistance metric consequences would follow immediately; if not, the class of schemes covered by Theorem 1.12 may exclude the most natural graph-based resistance approximations and the theorem would not apply to them.
  • The superpolynomial crossing decay proved here is likely a general feature of any CLE$_{\kappa'}$ metric in the axiom class, which would give a universal scaling relation between the metric and Euclidean diameter and could be tested numerically on critical percolation clusters.
  • The GHf topology with prime ends is the natural framework for any random fractal metric whose topology is not Euclidean; the same construction could be applied to supercritical LQG or CLE8 where similar non-Euclidean limits are expected.
  • A concrete check of the percolation connection: the established exponent bounds for chemical distance in critical percolation ($\alpha>1$ and $\alpha<4/3$) should match the scaling exponent of the geodesic CLE$_6$ metric; computing that metric's exponent is a by-product of this program.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces an axiomatic class of approximate CLE_kappa' metrics in the non-simple regime kappa' in (4,8), proves tightness of the renormalized internal metrics m_epsilon^{-1} d_epsilon under a set of hypotheses called good approximation schemes, and shows that subsequential limits satisfy the axioms of a CLE metric. The proof strategy is a bootstrap on an intersection crossing exponent: an a priori bound with exponent d_dbl is improved to a superpolynomial tail via a detailed analysis of individual bubbles, then chaining arguments yield tightness and Hölder-type estimates. Additional results include a zero/positive dichotomy for CLE metrics, tightness across scales, and a statement that geodesic approximation schemes have non-degenerate geodesic limits.

Significance. If the main results are correct, the paper provides a general and modular framework for constructing conformally covariant metrics on CLE gaskets in the intersecting regime, which is a key missing ingredient for the percolation chemical distance and resistance metric programs. The conditional theorems are supported by a 137-page proof with explicit hypotheses; the bootstrap from the a priori exponent d_dbl to arbitrary exponents is structurally sound and does not appear circular, since the median normalization is defined from the metric itself and the small-scale estimates are proved rather than assumed. The authors are also commendably explicit that the resistance-metric verification is deferred to [MY25a] and that non-degeneracy is not proved in full generality for arbitrary good approximation schemes. The main weakness is that the normalization used in the good-scheme hypothesis is presented ambiguously, and the abstract overstates the non-degeneracy statement for subsequential limits.

major comments (3)
  1. [Section 1.4, Definition 1.6, Eq. (1.4)] The good-scheme hypothesis is not fully specified as written. The text introduces a heuristic normalizer \hat m_epsilon based on the event E, says it will turn out that the two definitions are comparable, and then writes Eq. (1.4) as lim a_epsilon/(epsilon^{a0} m_epsilon)=0 using m_epsilon without the hat. If Eq. (1.4) is meant to refer to the Section 3.1 quantile, then the definition relies on a forward reference and the promised comparability with \hat m_epsilon is never proved; if it is meant to refer to \hat m_epsilon, then the hypothesis does not control the normalizer used throughout Section 3 and in the proof of Theorem 1.12. Either way the hypotheses of the main tightness theorem need to be stated rigorously, either by moving the definition of m_epsilon before Definition 1.6 or by proving the asserted comparability.
  2. [Abstract and Section 1.4 (after Theorem 1.13)] The abstract claims that every subsequential limit is a non-trivial metric on the CLE gasket, but the text explicitly disclaims a general non-degeneracy criterion: the paragraph after Theorem 1.13 says the authors have decided not to give a general criterion ensuring the limit is not identically zero, and only Theorem 1.17 proves non-degeneracy for geodesic approximation schemes. This overstates the unconditional content of Theorem 1.12 and Theorem 1.13. Please temper the abstract and introduction, or add a non-degeneracy theorem for the full class of good approximation schemes.
  3. [Section 1.2.1 and Section 6.3] The manuscript presents two advertised classes of examples, geodesic and resistance approximations, but the resistance verification is deferred to [MY25a] and the main theorems are conditional on axioms that are tailored with compatibility and monotonicity restrictions for resistance-type metrics. The geodesic examples are verified in Section 6.3, but the abstract and introduction should state more clearly that the unconditional results cover the geodesic schemes, while the resistance metric is a conditional example pending [MY25a].
minor comments (4)
  1. [Theorem 1.12] The statement says 'there exists a family of normalizing constants m_epsilon > 0', but m_epsilon is already part of the good-scheme setup via Definition 1.6 and Section 3.1. This phrasing suggests the constants are part of the conclusion; please rephrase to avoid an apparent circularity.
  2. [Section 1.4] The heuristic normalizer \hat m_epsilon is defined carefully but is never used again after Definition 1.6. If it is only motivational, say so explicitly; if it is used in later verification of examples, the missing comparability proof should be supplied or referenced.
  3. [Section 3.1] The definition of m_epsilon as a quantile of a supremum over X^int_{\delta,\epsilon} is stated for the specific two-flow-line setup but is used in Definition 1.6 for an arbitrary approximate CLE metric. Please clarify that the Section 3.1 construction yields a well-defined functional of the approximate CLE metric for the canonical pair of intersecting loops, and state where finiteness and independence of auxiliary choices are proved.
  4. [Throughout] The paper relies heavily on companion papers [AMY25], [MY25a], and [MY25b]. It would improve readability to add a short table or paragraph at the start of each technical section listing which results are imported and which are new.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the tightness proof is a genuine bootstrap from the axioms, with the median normalization defined from the metric but the multi-scale estimates proved rather than assumed.

full rationale

I walked the derivation chain: Definition 1.5 (axioms) + Definition 1.6 (good scheme) + external SLE/CLE inputs (d_SLE, d_dbl from [RS05], [MW17]; CLE resampling from [AMY25]) feed Proposition 3.1, whose proof is a self-consistency bootstrap, not a reduction. The normalizing constant m_epsilon (Section 3.1, q_epsilon(1/2) of the sup of internal distances over X_int_{δ,ε}) is defined from the metric, but the content of Proposition 3.1 — that distances across smaller regions exceed m_epsilon only with superpolynomially small probability — is proved: the paper argues by contradiction that if the a priori bound (1.9) failed, the series law plus independence of subregions would force P[d(x0,y0; Γ) ≥ m_epsilon] to exceed 1/2, contradicting the quantile definition ('...contradicting the definition of m_epsilon', Section 1.6). That is a legitimate one-way use of the definition, not a circular prediction. Lemma 3.8 likewise derives scale-delta quantile bounds from the fixed macroscopic median, again by contradiction with the definition of q_epsilon(q), not by assuming it. I could not exhibit any equation where a stated output equals a stated input by construction, nor any fitted parameter renamed as a prediction. The genuine issues are gaps, which I flag per the review rule: (i) Definition 1.6 condition (1.4) is written with m_epsilon before that object is rigorously defined in Section 3.1, and the promised comparability with the heuristic ̂m_epsilon ('it will turn out that the two definitions are comparable for good approximation schemes', Sections 1.4 and 3.1) is asserted, never proved; this makes the hypothesis of Theorem 1.12 under-specified as written, but it is a missing proof, not a reduction of the thesis to the hypothesis. (ii) The paper explicitly defers verification that resistance-type approximations satisfy its axioms ('We will explain in [MY25a] that for certain graph approximations...', Section 1.2.1), so the headline class of examples is conditional on companion work; again an omission, not circularity. (iii) [AMY25] resampling and total-variation continuity results are load-bearing in the machinery (Lemmas 2.7, 2.9–2.11, 2.19, Proposition 2.13), but they concern the CLE loop ensemble itself, not the target metric statements, so per the hard rules this self-citation is independent evidence and does not raise the circularity score.

Assumptions & free parameters 4 free parameters · 7 assumptions · 1 invented entities

The paper's contribution is conditional: it proves tightness and limit-axiom preservation for any member of an axiomatically defined class, not for the specific resistance construction advertised in the title and abstract. The normalization m_epsilon is a quantile of the very metric being scaled. The axiom system plus [AMY25] inputs carry most of the weight; no numbers are fitted to data, and the external exponents d_dbl and d_SLE come from prior literature ([MW17], [RS05]).

free parameters (4)
  • m_epsilon (median normalization)
    Median (1/2-quantile) of the sup of d_epsilon distances across a region bounded between two intersecting CLE_kappa' loops, conditioned on the intersection event E (Definition 1.6 simplified, Section 3.1 precise).
  • cs >= 0 (Euclidean interaction scale)
    Input constant in compatibility, monotonicity, series law, and generalized parallel law (Definition 1.5); the axioms hold only at Euclidean scales at least cs times epsilon. Its value is unspecified; the theorems are uniform in cs.
  • cp(N) > 0 (parallel-law constant) = 1 for geodesic, N for resistance
    Constant in the generalized parallel law; the main results need only the V_x = V version (Remark 1.4); the resistance case is deferred to [MY25a].
  • a_epsilon, a_0 (approximation error scales) = example-dependent (e.g., 4pi*epsilon^2, 1)
    Good-scheme conditions (1.4)-(1.5): a_epsilon/epsilon^{a_0} m_epsilon -> 0 and small-scale distances bounded by a_epsilon with high probability; these carry the epsilon-scale error into the tightness statement.
assumptions (7)
  • standard math SLE basics: continuity, simple/self-intersecting/space-filling trichotomy, Hausdorff dimension min(1+kappa/8, 2) (Rohde-Schramm, Beffara), p-variation estimates (HY22)
    Section 2.2; background used throughout the paper.
  • standard math Conformal mapping estimates: Koebe distortion and Beurling estimate (Lemmas 2.1, 2.2)
    Section 2.1; used in pocket arguments and dimension transfers.
  • domain assumption CLE_kappa' existence and structure via BCLE_kappa'(0) exploration tree (MSW17), local finiteness, nested CLE, thin gasket estimates (Lemma 2.5, C.18)
    Section 2.3; defines the objects on which the metrics live.
  • domain assumption GFF couplings: flow lines, counterflow lines, duality, space-filling SLE_kappa', natural parameterization (MS16a-c, MS17, Zha19, RZ17)
    Section 2.4; the engine for the flow-line constructions of Sections 3-5.
  • domain assumption Resampling and multichordal CLE results from [AMY25] (Theorem 2.6, Lemmas 2.7-2.11, Proposition 2.13, Lemma 4.4)
    Same-group companion preprint; used to compare linking patterns at intersection points and for total-variation continuity of conditional laws; load-bearing and not verifiable within this review.
  • ad hoc to paper Axiom system for approximate CLE_kappa' metrics (Definition 1.5): separability (1.2), Markovian property, translation invariance, compatibility, monotonicity, series law, generalized parallel law
    The postulates under which all main theorems are proved; verification for resistance metrics is deferred to [MY25a].
  • ad hoc to paper Good approximation scheme conditions (Definition 1.6): (1.4) a_epsilon/(epsilon^{a_0} m_epsilon) -> 0, and (1.5) small-scale distances bounded by a_epsilon with probability tending to 1
    Described as 'technical but rather mild' and 'satisfied for all reasonable approximation schemes'; not verified generally in this paper.
invented entities (1)
  • Abstract class of approximate CLE_kappa' metrics (Definition 1.5)
    purpose: Frames the tightness theorem: every limit of renormalized approximations is again a metric in the class, with epsilon = 0.
    The axioms are postulates tailored to the proof; the only concretely verified members are geodesic schemes (Section 6.3); the resistance member is deferred to [MY25a], and the percolation interpretation is explicitly conjectural.

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Pith. "Pith review of Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets." pith.science (2026). https://pith.science/paper/FXKIILFB

@misc{pith2026250715589,
  author       = {Pith},
  title        = {Pith review of: Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXKIILFB}},
  note         = {Machine review of arXiv:2507.15589}
}
abstract

We study a class of approximation schemes aimed at constructing conformally covariant metrics defined in the gasket of a conformal loop ensemble (CLE$_\kappa$) for $\kappa \in (4,8)$. This is the range of parameter values so that the loops of a CLE$_\kappa$ intersect themselves, each other, and the domain boundary. Its gasket is the closure of the union of the set of points not surrounded by a loop. The class of approximation schemes includes approximations to the geodesic metric and to the resistance metric. We show that the laws of these approximations are tight, and that every subsequential limit is a non-trivial metric on the CLE$_\kappa$ gasket satisfying a natural list of properties. Subsequent work of the second two authors will show that the limits exist and are conformally covariant both in the setting of the geodesic and resistance metrics. We conjecture that the geodesic (resp. resistance) metric describes the scaling limit of the chemical distance (resp. resistance) metric associated with discrete models that converge in the limit to CLE$_\kappa$ for $\kappa \in (4,8)$ (e.g., critical percolation for $\kappa=6$).

Figures

Figures reproduced from arXiv: 2507.15589 by the authors.

Figure 1.1
Figure 1.1. Top left: Critical percolation configuration generated on the hexagonal lattice in the unit disk. Top right: Regions surrounded by the outermost cluster that surrounds the origin. Bottom: The corresponding cluster of open sites, colored according to their graph distance to a point (shown in red) near the center [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Left: Illustration of the compatibility axiom. Shown is an example of regions V ⊆ V ′ (V is shaded in yellow, and V ′ is tiled in grey) where V ′\V is contained in a “dead end” which is separated by the point u shown in red. The compatibility axiom states that in the region below the red dot we have d V ϵ (·, ·; Γ) = d V ′ ϵ (·, ·; Γ). Right: Illustration of the extra condition in the monotonicity axiom. Shown is an… view at source ↗
Figure 1.3
Figure 1.3. Left: Illustration of the series law. It holds under the extra condition that the two yellow regions have Euclidean distance at least csϵ from each other. The region V is tiled in grey. Right: Illustration of the generalized parallel law with N = 3. The region V is tiled in grey, the points z1, ..., zN are shown in blue. The region Vx needs to contain a csϵ-neighborhood of Kx in V (shaded in yellow). that we may nee… view at source ↗
Figures from the paper (23 more)
Figure 1.4
Figure 1.4. Figure 1.4: We renormalize the approximate CLEκ′ metrics by the median of the distance across the region between two typical intersecting CLEκ′ loops as illustrated in the figure. The loops L1,L2 are shown in green and blue, respectively, and the region Ux′ ,y′ is shown in yello…
Figure 1.5
Figure 1.5. Figure 1.5: Left: A simplified illustration of a few intersecting CLEκ′ loops (only their outer boundaries are shown). Right: By resampling the CLEκ′ near the points marked with an ‘x’, we can compare the region between the two red points to a region that is bounded between two …
Figure 1.6
Figure 1.6. Figure 1.6: Left: The self-similar structure of the regions bounded between two intersecting CLEκ′ loops. Right: Inside each bubble, we find the same self-similar structure on both the left and the right side (only the outer boundaries of the CLEκ′ loops are shown) [PITH_FULL_I…
Figure 1.7
Figure 1.7. Figure 1.7: By resampling the CLEκ′ near the points marked with an ‘x’, we can compare the regions bounded by a finite collection of loops to the type of regions bounded by a single loop. In order to argue that we have sufficient independence so that all of this works, we will n…
Figure 2.1
Figure 2.1. Figure 2.1: By the duality, each component disconnected by a counterflow line can be detected by a pair of flow lines where one flow line is reflected off the other in the opposite direction. separated by a counterflow line using flow lines which we now explain. See [PITH_FULL_…
Figure 2.2
Figure 2.2. Figure 2.2: Illustration of the set Xθ z,r and a flow line η θ u with angle θ starting from u ∈ B(z, r/4). Then the set ∂A(z, r/4, 3r/4)∩ Xθ z,r is almost surely finite. Indeed, this follows from the almost sure continuity of space-filling SLEκ′ (since each distinct point on ∂A(…
Figure 3.1
Figure 3.1. Figure 3.1: The setup of Section 3. which by scaling does not depend on δ. For x, y ∈ η δ 1 ∩ η δ 2 ∩ D, let Ux,y denote the region bounded between the segments of η δ 1 , ηδ 2 from x to y. Let X int δ be the set of pairs (x, y) ∈ (η δ 1 ∩ η δ 2 ) 2 such that Ux,y ⊆ B(0, 3δ/4). …
Figure 3.2
Figure 3.2. Figure 3.2: The setup of Lemma 3.7. On the event G0,δ, the conditional probability is at least p that the dashed flow lines exit δD at iδ. Lemma 3.7. Fix q > 0, M, p > 0. Let w1, w2 ∈ B(0, δ/4), and let E denote the event that the following hold. • Let η θ1+π w1 , η θ1 w1 (resp.…
Figure 4.1
Figure 4.1. Figure 4.1: The setup of the main result of Section 4. Note that the right outer boundary η R of η ′ intersects η0 if and only if κ ′ < 6. Let δ > 0, and let D ⊆ C be a simply connected domain, x0, y0 ∈ ∂D with dist(x0, y0) < δ. All results in this section will be uniform in the…
Figure 4.2
Figure 4.2. Figure 4.2: The setup described in Lemma 4.7 for the intersections with η0. The light green loop represents the outer boundary of L. Shown are the three cases in the proof of the lemma. In each case, an angle −π/2 flow line merges into the part of the boundary of L that intersec…
Figure 4.3
Figure 4.3. Figure 4.3: The setup described in Lemma 4.8 for the intersections with η2. The light green loop represents the outer boundary of L. Shown are the three cases in the proof of the lemma. Lemma 4.7. Let z ∈ D and r > 0 such that 2r < δ1+a3 and B(z, r) ⊆ D. Let e ∈ Eδ 0 (L) for som…
Figure 4.4
Figure 4.4. Figure 4.4: Illustration of the setup of Lemma 4.10 (left) and 4.11 (right). Note that modulo 2π, the flow lines η R w1 (resp. η R) and η θ0 z1 have the same angle. Rather than merging into η θ0 z1 when hitting it, η R w1 (resp. η R) reflects off η θ0 z1 in the opposite directio…
Figure 4.5
Figure 4.5. Figure 4.5: Illustration of the setup of Lemma 4.13 (left) and 4.14 (right). The dashed curves occur on the events described in the last bullet points. In Lemma 4.14 (right), the behavior of the flow lines η R w1 , η R ensure that we draw a loop that exits B(0, 3δ/4). We now con…
Figure 4.6
Figure 4.6. Figure 4.6: Illustration of the concatenation argument used in the proofs of Sec￾tion 4.3. If the distance between the blue points is bounded, then by the generalized parallel law for at least one of the red points the distance across the right grey arrow is also bounded (up to …
Figure 4.7
Figure 4.7. Figure 4.7: Creating an event on which the CLE configuration is locally determined by the GFF. such Am there is a crosscut Ξ ⊆ Am connecting [−2 m, −2 m−1 ] to [2m, 2 m−1 ] such that ψ −1 (Ξ) ⊆ A(0, δ1+3a1 , δ1+2a1 ). It follows that (on the event E1 δ ) we have ψ −1 (Am) ⊆ A(0,…
Figure 4.8
Figure 4.8. Figure 4.8: The CLE configuration in the regions bounded between the dark green loops can be recovered from the values of a GFF in B(0, δ1+a1 ). Let h be a GFF generating (η, Γ sep) in such a way that η is the flow line with angle −θ0 = −3π/2 and Γ sep is constructed from the br…
Figure 4.9
Figure 4.9. Figure 4.9: Shown in pink and blue is one pair of ηe θ1 i1,i2 , ηe θ2 i1,i2 in the definition of the event E4 j [PITH_FULL_IMAGE:figures/full_fig_p085_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: Loops in Γe+ are shown in black, loops in Γb are shown in dark green. Left: Shown in purple is a pair zi1 , zi2 with (i1, i2) ∈ I. We will link the loops at the red points. Right: The orange points are the set {zi : i ∈ I ′′}. Let (zi) and K be as in the definition …
Figure 4.11
Figure 4.11. Figure 4.11: In the proof of Proposition 4.1, we distinguish between the various parts of η2 which are shown in different shades of blue. To summarize, (on E4) each point of η2 satisfies at least one of the following (see [PITH_FULL_IMAGE:figures/full_fig_p091_4_11.png]
Figure 5.1
Figure 5.1. Figure 5.1: The setup of Proposition 5.4. between v and y. Let Ux,y be the region bounded between the segments of ηu,v and ηu from x to y. Let Γx,y be the collection of conditionally independent CLEκ′ naturally coupled with h in each of the components of Ux,y. Let ΥΓx,y denote t…
Figure 5.2
Figure 5.2. Figure 5.2: The setup and proof of Lemma 5.11. If the right side of ηbu,v does not have too small harmonic measure, then there is a sufficiently high chance that the bubble can be detected and compared to the setup of Lemma 5.8. We split the proof of Lemma 5.10 into several step…
Figure 5.3
Figure 5.3. Figure 5.3: Several atypical shapes of bubbles that are ruled out in Lemma 5.12. The first two scenarios are ruled out in the event F3, the third scenario would create an approximate double point and is therefore unlikely. Lemma 5.12. Consider the setup of Lemma 5.11. For any c2…
Figure 5.4
Figure 5.4. Figure 5.4: An unlikely event in the proof of Lemma 5.10. Proof of Lemma 5.10. Recall the setup of Proposition 5.4. We can assume that diam(Ux,y) ∈ [δ/2, δ] and |u − x| ≥ δ/4. Fix b > 0. Let G1 be the event that the event in Lemma 5.13 does not occur for any δ ′ < δ. We choose c…

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Reference graph

Works this paper leans on

98 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aizenman and A

    M. Aizenman and A. Burchard. H\" o lder regularity and dimension bounds for random curves. Duke Math. J. , 99(3):419--453, 1999

    1999

  2. [2]

    Scaling limit for the random walk on critical lattice trees

    G\'erard Ben Arous, Manuel Cabezas, and Alexander Fribergh. Scaling limit for the random walk on critical lattice trees. ArXiv e-prints , 2025

    2025

  3. [3]

    A continuous proof of the existence of the SLE _8 curve

    Valeria Ambrosio and Jason Miller . A continuous proof of the existence of the SLE _8 curve . arXiv e-prints , page arXiv:2203.13805, March 2022

    arXiv 2022

  4. [4]

    Multiple SLE _ from CLE _ for (4,8)

    Valeria Ambrosio, Jason Miller, and Yizheng Yuan. Multiple SLE _ from CLE _ for (4,8) . ArXiv e-prints , 2025

    2025

  5. [5]

    On the chemical distance for supercritical B ernoulli percolation

    Peter Antal and Agoston Pisztora. On the chemical distance for supercritical B ernoulli percolation. Ann. Probab. , 24(2):1036--1048, 1996

    1996

  6. [6]

    Martin T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. , 32(4):3024--3084, 2004

    2004

  7. [7]

    Quenched invariance principle for simple random walk on percolation clusters

    Noam Berger and Marek Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields , 137(1-2):83--120, 2007

    2007

  8. [8]

    A course in metric geometry , volume 33 of Graduate Studies in Mathematics

    Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry , volume 33 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2001

    2001

Show all 98 references
  1. [9]

    M. T. Barlow, D. A. Croydon, and T. Kumagai. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Ann. Probab. , 45(1):4--55, 2017

    2017

  2. [10]

    The dimension of the SLE curves

    Vincent Beffara. The dimension of the SLE curves. Ann. Probab. , 36(4):1421--1452, 2008

    2008

  3. [11]

    The scaling limit of critical I sing interfaces is CLE_3

    St\' e phane Benoist and Cl\' e ment Hongler. The scaling limit of critical I sing interfaces is CLE_3 . Ann. Probab. , 47(4):2049--2086, 2019

    2019

  4. [12]

    Barlow, Antal A

    Martin T. Barlow, Antal A. J\'arai, Takashi Kumagai, and Gordon Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys. , 278(2):385--431, 2008

    2008

  5. [13]

    The scaling limit of critical hypercube percolation

    Arthur Blanc-Renaudie, Nicolas Broutin, and Asaf Nachmias. The scaling limit of critical hypercube percolation. ArXiv e-prints , 2024

    2024

  6. [14]

    The scaling limit of planar maps with large faces

    Nicholas Curien, Gr\' e gory Miermont, and Armand Riera. The scaling limit of planar maps with large faces. ArXiv e-prints , 2025

    2025

  7. [15]

    Federico Camia and Charles M. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys. , 268(1):1--38, 2006

    2006

  8. [16]

    Tightness of L iouville first passage percolation for (0,2)

    Jian Ding, Julien Dub\' e dat, Alexander Dunlap, and Hugo Falconet. Tightness of L iouville first passage percolation for (0,2) . Publ. Math. Inst. Hautes \' E tudes Sci. , 132:353--403, 2020

    2020

  9. [17]

    Tightness of supercritical L iouville first passage percolation

    Jian Ding and Ewain Gwynne. Tightness of supercritical L iouville first passage percolation. J. Eur. Math. Soc. (JEMS) , 25(10):3833--3911, 2023

    2023

  10. [18]

    Uniqueness of the critical and supercritical L iouville quantum gravity metrics

    Jian Ding and Ewain Gwynne. Uniqueness of the critical and supercritical L iouville quantum gravity metrics. Proc. Lond. Math. Soc. (3) , 126(1):216--333, 2023

    2023

  11. [19]

    On the chemical distance in critical percolation

    Michael Damron, Jack Hanson, and Philippe Sosoe. On the chemical distance in critical percolation. Electron. J. Probab. , 22:Paper No. 75, 43, 2017

    2017

  12. [20]

    Strict inequality for the chemical distance exponent in two-dimensional critical percolation

    Michael Damron, Jack Hanson, and Philippe Sosoe. Strict inequality for the chemical distance exponent in two-dimensional critical percolation. Comm. Pure Appl. Math. , 74(4):679--743, 2021

    2021

  13. [21]

    Connectivity of the adjacency graph of complementary components of the SLE fan

    Cillian Doherty, Konstantinos Kavvadias, and Jason Miller. Connectivity of the adjacency graph of complementary components of the SLE fan. ArXiv e-prints , 2024

    2024

  14. [22]

    Convergence of the intrinsic metric and simple random walk on two-dimensional critical percolation clusters

    Irina D ankovi\' c , Maarten Markering, Jason Miller, and Yizheng Yuan. Convergence of the intrinsic metric and simple random walk on two-dimensional critical percolation clusters. 2025. In preparation

    2025

  15. [23]

    Liouville quantum gravity as a mating of trees

    Bertrand Duplantier, Jason Miller, and Scott Sheffield. Liouville quantum gravity as a mating of trees. Ast\' e risque , (427):viii+257, 2021

    2021

  16. [24]

    S LE and the free field: partition functions and couplings

    Julien Dub\' e dat. S LE and the free field: partition functions and couplings. J. Amer. Math. Soc. , 22(4):995--1054, 2009

    2009

  17. [25]

    Laurence S. Field. Two-sided radial SLE and length-biased chordal SLE . ArXiv e-prints , 2016

    2016

  18. [26]

    Field and Gregory F

    Laurence S. Field and Gregory F. Lawler. Escape probability and transience for SLE . Electron. J. Probab. , 20:no. 10, 14, 2015

    2015

  19. [27]

    An almost sure KPZ relation for SLE and B rownian motion

    Ewain Gwynne, Nina Holden, and Jason Miller. An almost sure KPZ relation for SLE and B rownian motion. Ann. Probab. , 48(2):527--573, 2020

    2020

  20. [28]

    Ewain Gwynne, Adrien Kassel, Jason Miller, and David B. Wilson. Active spanning trees with bending energy on planar maps and SLE -decorated L iouville quantum gravity for > 8 . Comm. Math. Phys. , 358(3):1065--1115, 2018

    2018

  21. [29]

    Shirshendu Ganguly and James R. Lee. Chemical subdiffusivity of critical 2 D percolation. Comm. Math. Phys. , 389(2):695--714, 2022

    2022

  22. [30]

    G\" a rtner and S

    J. G\" a rtner and S. A. Molchanov. Parabolic problems for the A nderson model. I . I ntermittency and related topics. Comm. Math. Phys. , 132(3):613--655, 1990

    1990

  23. [31]

    G. R. Grimmett and J. M. Marstrand. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A , 430(1879):439--457, 1990

    1990

  24. [32]

    Large deviations for the chemical distance in supercritical B ernoulli percolation

    Olivier Garet and R\' e gine Marchand. Large deviations for the chemical distance in supercritical B ernoulli percolation. Ann. Probab. , 35(3):833--866, 2007

    2007

  25. [33]

    Scaling limit of the uniform infinite half-plane quadrangulation in the G romov- H ausdorff- P rokhorov-uniform topology

    Ewain Gwynne and Jason Miller. Scaling limit of the uniform infinite half-plane quadrangulation in the G romov- H ausdorff- P rokhorov-uniform topology. Electron. J. Probab. , 22:Paper No. 84, 47, 2017

    2017

  26. [34]

    Convergence of the self-avoiding walk on random quadrangulations to SLE_ 8/3 on 8/3 - L iouville quantum gravity

    Ewain Gwynne and Jason Miller. Convergence of the self-avoiding walk on random quadrangulations to SLE_ 8/3 on 8/3 - L iouville quantum gravity. Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 54(2):305--405, 2021

    2021

  27. [35]

    Existence and uniqueness of the L iouville quantum gravity metric for (0,2)

    Ewain Gwynne and Jason Miller. Existence and uniqueness of the L iouville quantum gravity metric for (0,2) . Invent. Math. , 223(1):213--333, 2021

    2021

  28. [36]

    Percolation on uniform quadrangulations and SLE_6 on 8/3 - L iouville quantum gravity

    Ewain Gwynne and Jason Miller. Percolation on uniform quadrangulations and SLE_6 on 8/3 - L iouville quantum gravity. Ast\' e risque , (429):vii+242, 2021

    2021

  29. [37]

    Conformal invariance of CLE _ on the R iemann sphere for (4, 8)

    Ewain Gwynne, Jason Miller, and Wei Qian. Conformal invariance of CLE _ on the R iemann sphere for (4, 8) . Int. Math. Res. Not. IMRN , (23):17971--18036, 2021

    2021

  30. [38]

    Convergence in distribution of random metric measure spaces ( -coalescent measure trees)

    Andreas Greven, Peter Pfaffelhuber, and Anita Winter. Convergence in distribution of random metric measure spaces ( -coalescent measure trees). Probab. Theory Related Fields , 145(1-2):285--322, 2009

    2009

  31. [39]

    Herrmann and H

    Hans J. Herrmann and H. Eugene Stanley. The fractal dimension of the minimum path in two- and three-dimensional percolation. J. Phys. A , 21(17):L829--L833, 1988

    1988

  32. [40]

    S LE as a mating of trees in E uclidean geometry

    Nina Holden and Xin Sun. S LE as a mating of trees in E uclidean geometry. Comm. Math. Phys. , 364(1):171--201, 2018

    2018

  33. [41]

    Barry D. Hughes. Random walks and random environments. V ol. 2 . Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. Random environments

    1996

  34. [42]

    Random walk on the high-dimensional IIC

    Markus Heydenreich, Remco van der Hofstad, and Tim Hulshof. Random walk on the high-dimensional IIC . Comm. Math. Phys. , 329(1):57--115, 2014

    2014

  35. [43]

    Regularity of the S chramm- L oewner evolution: U p-to-constant variation and modulus of continuity

    Nina Holden and Yizheng Yuan. Regularity of the S chramm- L oewner evolution: U p-to-constant variation and modulus of continuity. ArXiv e-prints , 2022

    2022

  36. [44]

    Subdiffusive behavior of random walk on a random cluster

    Harry Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincar\'e Probab. Statist. , 22(4):425--487, 1986

    1986

  37. [45]

    Richard Kenyon, Jason Miller, Scott Sheffield, and David B. Wilson. Bipolar orientations on planar maps and SLE _ 12 . Ann. Probab. , 47(3):1240--1269, 2019

    2019

  38. [46]

    The A lexander- O rbach conjecture holds in high dimensions

    Gady Kozma and Asaf Nachmias. The A lexander- O rbach conjecture holds in high dimensions. Invent. Math. , 178(3):635--654, 2009

    2009

  39. [47]

    Conformal invariance of boundary touching loops of FK I sing model

    Antti Kemppainen and Stanislav Smirnov. Conformal invariance of boundary touching loops of FK I sing model. Comm. Math. Phys. , 369(1):49--98, 2019

    2019

  40. [48]

    The tortuosity of occupied crossings of a box in critical percolation

    Harry Kesten and Yu Zhang. The tortuosity of occupied crossings of a box in critical percolation. J. Statist. Phys. , 70(3-4):599--611, 1993

    1993

  41. [49]

    Gregory F. Lawler. Conformally invariant processes in the plane , volume 114 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2005

    2005

  42. [50]

    Uniqueness and universality of the B rownian map

    Jean-Fran c ois Le Gall. Uniqueness and universality of the B rownian map. Ann. Probab. , 41(4):2880--2960, 2013

    2013

  43. [51]

    Scaling limits of random planar maps with large faces

    Jean-Fran c ois Le Gall and Gr\' e gory Miermont. Scaling limits of random planar maps with large faces. Ann. Probab. , 39(1):1--69, 2011

    2011

  44. [52]

    Conformal restriction: the chordal case

    Gregory Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. , 16(4):917--955, 2003

    2003

  45. [53]

    Lawler, Oded Schramm, and Wendelin Werner

    Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. , 32(1B):939--995, 2004

    2004

  46. [54]

    Yiting Li , Xin Sun , and Samuel S. Watson . Schnyder woods, SLE(16), and Liouville quantum gravity . arXiv e-prints , page arXiv:1705.03573, May 2017

    2017 arXiv

  47. [55]

    Lawler and Fredrik Viklund

    Gregory F. Lawler and Fredrik Viklund. Convergence of loop-erased random walk in the natural parameterization. Duke Math. J. , 170(10):2289--2370, 2021

    2021

  48. [56]

    The B rownian map is the scaling limit of uniform random plane quadrangulations

    Gr\' e gory Miermont. The B rownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. , 210(2):319--401, 2013

    2013

  49. [57]

    Tightness of approximations to the chemical distance metric for simple conformal loop ensembles

    Jason Miller . Tightness of approximations to the chemical distance metric for simple conformal loop ensembles . arXiv e-prints , page arXiv:2112.08335, December 2021

    2021 arXiv

  50. [58]

    Mathieu and A

    P. Mathieu and A. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 463(2085):2287--2307, 2007

    2007

  51. [59]

    The geodesics in L iouville quantum gravity are not S chramm- L oewner evolutions

    Jason Miller and Wei Qian. The geodesics in L iouville quantum gravity are not S chramm- L oewner evolutions. Probab. Theory Related Fields , 177(3-4):677--709, 2020

    2020

  52. [60]

    Imaginary geometry I : interacting SLE s

    Jason Miller and Scott Sheffield. Imaginary geometry I : interacting SLE s. Probab. Theory Related Fields , 164(3-4):553--705, 2016

    2016

  53. [61]

    Imaginary geometry II : reversibility of SLE _ ( _1; _2) for (0,4)

    Jason Miller and Scott Sheffield. Imaginary geometry II : reversibility of SLE _ ( _1; _2) for (0,4) . Ann. Probab. , 44(3):1647--1722, 2016

    2016

  54. [62]

    Imaginary geometry III : reversibility of SLE_ for (4,8)

    Jason Miller and Scott Sheffield. Imaginary geometry III : reversibility of SLE_ for (4,8) . Ann. of Math. (2) , 184(2):455--486, 2016

    2016

  55. [63]

    Imaginary geometry IV : interior rays, whole-plane reversibility, and space-filling trees

    Jason Miller and Scott Sheffield. Imaginary geometry IV : interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields , 169(3-4):729--869, 2017

    2017

  56. [64]

    Gaussian free field light cones and SLE _ ( )

    Jason Miller and Scott Sheffield. Gaussian free field light cones and SLE _ ( ) . Ann. Probab. , 47(6):3606--3648, 2019

    2019

  57. [65]

    Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric. Invent. Math. , 219(1):75--152, 2020

    2020

  58. [66]

    Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding. Ann. Probab. , 49(6):2732--2829, 2021

    2021

  59. [67]

    Liouville quantum gravity and the B rownian map III : the conformal structure is determined

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map III : the conformal structure is determined. Probab. Theory Related Fields , 179(3-4):1183--1211, 2021

    2021

  60. [68]

    Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

    Jason Miller and Lukas Schoug . Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles . arXiv e-prints , page arXiv:2201.01748, January 2022

    2022 arXiv

  61. [69]

    Jason Miller, Nike Sun, and David B. Wilson. The H ausdorff dimension of the CLE gasket. Ann. Probab. , 42(4):1644--1665, 2014

    2014

  62. [70]

    C LE percolations

    Jason Miller, Scott Sheffield, and Wendelin Werner. C LE percolations. Forum Math. Pi , 5:e4, 102, 2017

    2017

  63. [71]

    Intersections of SLE paths: the double and cut point dimension of SLE

    Jason Miller and Hao Wu. Intersections of SLE paths: the double and cut point dimension of SLE . Probab. Theory Related Fields , 167(1-2):45--105, 2017

    2017

  64. [72]

    Existence and uniqueness of the canonical diffusion on non-simple conformal loop ensemble gaskets

    Jason Miller and Yizheng Yuan. Existence and uniqueness of the canonical diffusion on non-simple conformal loop ensemble gaskets. 2025. In preparation

    2025

  65. [73]

    Existence and uniqueness of the conformally covariant metric on non-simple conformal loop ensemble gaskets

    Jason Miller and Yizheng Yuan. Existence and uniqueness of the conformally covariant metric on non-simple conformal loop ensemble gaskets. 2025. In preparation

    2025

  66. [74]

    Backbone exponent for two-dimensional percolation

    Pierre Nolin, Wei Qian, Xin Sun, and Zijie Zhuang. Backbone exponent for two-dimensional percolation. ArXiv e-prints , 2023

    2023

  67. [75]

    Random soups, carpets and fractal dimensions

    S erban Nacu and Wendelin Werner. Random soups, carpets and fractal dimensions. J. Lond. Math. Soc. (2) , 83(3):789--809, 2011

    2011

  68. [76]

    Shortest path and schramm-loewner evolution

    Nicolas Pos \'e , K Julian Schrenk, Nuno AM Ara \'u jo, and Hans J Herrmann. Shortest path and schramm-loewner evolution. Scientific reports , 4(1):5495, 2014

    2014

  69. [77]

    Basic properties of SLE

    Steffen Rohde and Oded Schramm. Basic properties of SLE . Ann. of Math. (2) , 161(2):883--924, 2005

    2005

  70. [78]

    Rezaei and Dapeng Zhan

    Mohammad A. Rezaei and Dapeng Zhan. Higher moments of the natural parameterization for SLE curves. Ann. Inst. Henri Poincar\' e Probab. Stat. , 53(1):182--199, 2017

    2017

  71. [79]

    Scaling limits of loop-erased random walks and uniform spanning trees

    Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. , 118:221--288, 2000

    2000

  72. [80]

    Conformally invariant scaling limits: an overview and a collection of problems

    Oded Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International C ongress of M athematicians. V ol. I , pages 513--543. Eur. Math. Soc., Z\" u rich, 2007

    2007

  73. [81]

    Gaussian free fields for mathematicians

    Scott Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields , 139(3-4):521--541, 2007

    2007

  74. [82]

    Exploration trees and conformal loop ensembles

    Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79--129, 2009

    2009

  75. [83]

    Conformal weldings of random surfaces: SLE and the quantum gravity zipper

    Scott Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. , 44(5):3474--3545, 2016

    2016

  76. [84]

    Quantum gravity and inventory accumulation

    Scott Sheffield. Quantum gravity and inventory accumulation. Ann. Probab. , 44(6):3804--3848, 2016

    2016

  77. [85]

    Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits

    Stanislav Smirnov. Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits. C. R. Acad. Sci. Paris S\' e r. I Math. , 333(3):239--244, 2001

    2001

  78. [86]

    Conformal invariance in random cluster models

    Stanislav Smirnov. Conformal invariance in random cluster models. I . H olomorphic fermions in the I sing model. Ann. of Math. (2) , 172(2):1435--1467, 2010

    2010

  79. [87]

    An estimate for the radial chemical distance in 2d critical percolation clusters

    Philippe Sosoe and Lily Reeves. An estimate for the radial chemical distance in 2d critical percolation clusters. Stochastic Process. Appl. , 147:145--174, 2022

    2022

  80. [88]

    Quenched invariance principles for walks on clusters of percolation or among random conductances

    Vladas Sidoravicius and Alain-Sol Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields , 129(2):219--244, 2004

    2004

  81. [89]

    Contour lines of the two-dimensional discrete G aussian free field

    Oded Schramm and Scott Sheffield. Contour lines of the two-dimensional discrete G aussian free field. Acta Math. , 202(1):21--137, 2009

    2009

  82. [90]

    A contour line of the continuum G aussian free field

    Oded Schramm and Scott Sheffield. A contour line of the continuum G aussian free field. Probab. Theory Related Fields , 157(1-2):47--80, 2013

    2013

  83. [91]

    Oded Schramm, Scott Sheffield, and David B. Wilson. Conformal radii for conformal loop ensembles. Comm. Math. Phys. , 288(1):43--53, 2009

    2009

  84. [92]

    Oded Schramm and David B. Wilson. S LE coordinate changes. New York J. Math. , 11:659--669, 2005

    2005

  85. [93]

    Conformal loop ensembles: the M arkovian characterization and the loop-soup construction

    Scott Sheffield and Wendelin Werner. Conformal loop ensembles: the M arkovian characterization and the loop-soup construction. Ann. of Math. (2) , 176(3):1827--1917, 2012

    1917

  86. [94]

    Boundary proximity of SLE

    Oded Schramm and Wang Zhou. Boundary proximity of SLE . Probab. Theory Related Fields , 146(3-4):435--450, 2010

    2010

  87. [95]

    Cycle structure of percolation on high-dimensional tori

    Remco van der Hofstad and Art\"em Sapozhnikov. Cycle structure of percolation on high-dimensional tori. Ann. Inst. Henri Poincar\'e Probab. Stat. , 50(3):999--1027, 2014

    2014

  88. [96]

    Alternating arm exponents for the critical planar I sing model

    Hao Wu. Alternating arm exponents for the critical planar I sing model. Ann. Probab. , 46(5):2863--2907, 2018

    2018

  89. [97]

    Decomposition of S chramm- L oewner evolution along its curve

    Dapeng Zhan. Decomposition of S chramm- L oewner evolution along its curve. Stochastic Process. Appl. , 129(1):129--152, 2019

    2019

  90. [98]

    S LE loop measures

    Dapeng Zhan. S LE loop measures. Probab. Theory Related Fields , 179(1-2):345--406, 2021

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.