arxiv: 2512.04807 · v2 · submitted 2025-12-04 · 🧮 math.PR · math-ph· math.CV· math.MP

Existence and uniqueness of the canonical Brownian motion in non-simple conformal loop ensemble gaskets

Jason Miller , Yizheng Yuan This is my paper

Pith reviewed 2026-05-17 01:24 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP

keywords conformal loop ensembleCLE gasketBrownian motionresistance formdiffusion processscaling limitrandom walkpercolation

0 comments

The pith

A unique diffusion process exists on the gasket of non-simple conformal loop ensembles for kappa between 4 and 8.

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

The paper shows there is a unique diffusion on the CLE_kappa gasket whose law is fixed locally by the loop structure and obeys translation invariance together with scale invariance up to time reparametrization. The same uniqueness holds for the associated resistance form, which must be locally determined by the CLE and scale-covariant. A reader would care because the result supplies a canonical continuous random motion on the random fractal sets produced by intersecting loops, providing a candidate for the scaling limit of lattice random walks in models that converge to these ensembles.

Core claim

The authors prove existence and uniqueness of the canonical Brownian motion on the CLE_kappa gasket for kappa in (4,8): any diffusion process whose law depends locally on the CLE and satisfies translation-invariance and scale-invariance modulo time change is this process, and likewise there exists a unique resistance form on the gasket that is locally determined by the CLE while being translation-invariant and scale-covariant.

What carries the argument

The resistance form on the CLE_kappa gasket, which determines the diffusion through its Dirichlet energy and is required to be locally fixed by the loop ensemble while obeying global translation invariance and scale covariance.

If this is right

The Brownian motion is conjectured to describe the scaling limit of simple random walks on two-dimensional statistical mechanics models that converge to CLE_kappa.
For kappa equals 6 the construction will be used to prove the scaling limit for critical percolation on the triangular lattice.
The resistance form supplies a canonical way to define effective resistances and conductances on these random gaskets.

Where Pith is reading between the lines

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

The same local-to-global uniqueness argument might be adapted to define heat kernels or Green functions directly from the resistance form on the gasket.
Numerical simulation of CLE realizations could test whether the effective resistance between distant points matches the form predicted by the paper.
The canonical diffusion could serve as a building block for studying other processes on random fractals that arise from intersecting loop models.

Load-bearing premise

Any candidate diffusion or resistance form must depend only locally on the CLE structure while obeying global translation and scale invariance; without this locality-plus-invariance requirement uniqueness can fail.

What would settle it

Constructing two distinct diffusions on the same CLE gasket that both depend locally on the loops and satisfy the stated invariance properties, or finding that random-walk scaling limits on a lattice model known to converge to CLE_kappa do not match the predicted process.

Figures

Figures reproduced from arXiv: 2512.04807 by Jason Miller, Yizheng Yuan.

**Figure 1.1.** Figure 1.1: Illustration of the locality property of the CLEκ′ resistance form. The energy of f is given by the sum of the energies of the restrictions of f to the subregions (indicated with different colors), and the energy of f in each subregion is locally determined by the CLEκ′. The subregions are connected at a finite number of points (indicated by red dots). III. If V, V ′ ∈ C, V ⊆ V ′ , and f : V ′ ∩ ΥΓ → R i… view at source ↗

**Figure 3.1.** Figure 3.1: Illustration of the locality property (vi) of weak CLEκ′ resistance forms. (A simplified sketch of) a loop L contained in V is shown in green. (iv) Suppose that V, V ′ ∈ C, V ⊆ V ′ , such that for every u ∈ V ′ \ V there is a point z ∈ V that separates u from V in V ′ ∩ ΥΓ. Let f : V ′ ∩ ΥΓ → R be a continuous function that is constant on each dpath-connected component of V ′ ∩ ΥΓ \ V . Then f ∈ FV ′ ,Γ … view at source ↗

**Figure 3.2.** Figure 3.2: Suppose V consists of the regions shaded in yellow and pink. If V ′ ⊇ V contains the green loop, then the pink region becomes part of V ′ (s(ϵ)), so V \ V(s(ϵ)) ⊈ V ′ \ V ′ (s(ϵ)). The dashed edges are also lost in GV ′ ϵ . more than one shortest path between v, w contained in Bv,w, then we draw two paths, the left-most one seen from v and from w. (We now explain that a left-most shortest path seen from … view at source ↗

**Figure 3.3.** Figure 3.3: Top left: The region U between the outer boundaries of η ′ 1 and η ′ 2 is shown in yellow. This region is used to define the renormalization constant mϵ. Top right: The additional dead ends in the region U ′ between η ′ 1 and η ′ 2 are shown in grey. Note that the cables emanating from the vertices in U ′ \ U create additional edges between the vertices in U. Bottom: The dead ends of size at most s(ϵ) ar… view at source ↗

**Figure 3.4.** Figure 3.4: Shown is a region V and some of the loops in V . The small dead ends in V(s(ϵ)) cut off by these loops are shown in pink. When we divide V into subcomponents (one is shown in yellow), we need to remove the dead ends also from the subcomponents in order to make their graph approximations match. We will however need a slight modification of the construction. Recall that we have proved the non-degeneracy in… view at source ↗

**Figure 4.1.** Figure 4.1: Illustration of the event E1 z,j , the points wi , vi , v′ i , ui , and the regions Vi , V ′ i . We let E2 z,j be the event that E1 z,j occurs and we can select the points so that additionally the following holds. For each i = 1, . . . , m, let Vi (resp. V ′ i ) be the region bounded between the segments of γ L i , γR i from vi to v ′ i (resp. v ′ i to ui). Then (4.2) RVi (vi , v′ i ; Γ), RV ′ i (v ′ i ,… view at source ↗

**Figure 4.2.** Figure 4.2: Continuation of [PITH_FULL_IMAGE:figures/full_fig_p047_4_2.png] view at source ↗

**Figure 4.3.** Figure 4.3: In the proof of Proposition 4.1, if Vl,k ∈ In−1 and vn,i ∈ Vl,k, it might be that gen−1(vn,j ) < infKn f. In the depicted case, we set gen to be constant in the region tiled in blue. Vl,k Ul Vl,k Ul Un V ′ l,k vn,i [PITH_FULL_IMAGE:figures/full_fig_p051_4_3.png] view at source ↗

**Figure 4.4.** Figure 4.4: In the proof of Proposition 4.1, if Vl,k ∈ In−1 and vn,i ∈ Vl,k, it might be that gen−1(vn,j ) < infKn f. In the depicted case, we harmonically interpolate in V ′ l,k (tiled in blue) instead of Vn,i. Proof of Proposition 4.1. Let M, r1, r2 be such that the conclusion of Corollary 4.11 holds. Let {Ul} be a minimal associated cover. We show that there is a constant c depending on M such that (4.6) eE V (·,… view at source ↗

read the original abstract

We construct the canonical Brownian motion on the gasket of conformal loop ensembles (CLE$_\kappa$) for $\kappa \in (4,8)$ (which is the range of parameter values in which loops of the CLE$_\kappa$ can intersect themselves, each other, and the domain boundary). More precisely, we show that there is a unique diffusion process on the CLE$_\kappa$ gasket whose law depends locally on the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-invariance (modulo time change). We characterize the diffusion process by its resistance form and show in particular that there is a unique resistance form on the CLE$_\kappa$ gasket that is locally determined by the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-covariance. We conjecture that the CLE$_\kappa$ Brownian motion describes the scaling limit of simple random walk on statistical mechanics models in two dimensions that converge to CLE$_\kappa$. In future work the results of this paper will be used to show that this is the case with $\kappa=6$ for critical percolation on the triangular lattice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a new existence-uniqueness theorem for a canonical diffusion on non-simple CLE_κ gaskets via resistance forms, but uniqueness depends on how strictly the locality condition is enforced.

read the letter

This paper constructs a unique diffusion on the gasket of non-simple CLE_κ for κ in (4,8) and characterizes it by a resistance form that is locally determined by the CLE while satisfying translation invariance and scale covariance (modulo time change). The main advance is the explicit construction and uniqueness proof in the intersecting-loop regime, which had not been settled in the cited prior work. They tie the object to conjectures on scaling limits of random walks on models that converge to CLE_κ and flag a concrete next step for critical percolation at κ=6. That linkage is direct and uses existing CLE properties rather than new ad-hoc assumptions. The resistance-form route is a natural fit for the fractal gasket and avoids obvious circularity by building from local CLE data plus standard invariance requirements. The argument appears to rest on established theory for resistance forms, which is a plus for reproducibility. The soft spot is the locality condition itself. If the definition of local dependence permits forms whose energy measures draw on averaged information from distant loops in a way that still respects the global invariances, then other candidates could satisfy the same axioms. The stress-test note flags exactly this point, and the paper needs to show that its precise formulation of locality rules out such alternatives with no residual freedom. Without that verification the uniqueness claim is less secure than the abstract suggests. This work is for people already following SLE/CLE and random geometry in the plane. A reader interested in diffusions on random fractals or scaling limits would find the construction and characterization useful. It is worth sending to a serious referee who can examine the technical estimates on locality and the resistance-form construction.

Referee Report

1 major / 1 minor

Summary. The paper constructs the canonical Brownian motion on the gasket of a conformal loop ensemble CLE_κ for κ ∈ (4,8). It proves existence and uniqueness of a diffusion process on this gasket whose law depends locally on the CLE_κ structure and satisfies translation-invariance together with scale-invariance (modulo time change). Equivalently, the diffusion is characterized by a unique resistance form on the gasket that is locally determined by the CLE_κ and obeys translation-invariance and scale-covariance. The result is conjectured to arise as the scaling limit of simple random walks on lattice models converging to CLE_κ, with a planned application to critical percolation on the triangular lattice for κ=6.

Significance. If the central claims hold, the work supplies a rigorous, canonical construction of Brownian motion on the non-simple CLE gaskets, which are fractal sets of interest in two-dimensional critical phenomena. The resistance-form characterization is a clear strength: it yields a parameter-free description once locality and the stated invariances are imposed, building directly on existing CLE properties and resistance-form theory. The explicit conjecture and forward reference to percolation scaling limits indicate a concrete path toward applications in statistical mechanics.

major comments (1)

[Main theorem / characterization of the resistance form] The uniqueness statement for the resistance form (central claim in the abstract and main theorem) rests on the precise meaning of 'locally determined by the CLE_κ'. The manuscript must verify that this locality condition excludes candidate forms whose local energy measures incorporate averaged information from distant loops while still obeying translation- and scale-invariance. Without an explicit argument or estimate showing that no such residual freedom remains, the uniqueness conclusion is not yet load-bearing.

minor comments (1)

[Introduction / statement of main result] Notation for the time-change in the scale-invariance statement should be made fully explicit when the diffusion is introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the explicit verification of the locality condition in our uniqueness argument. We address the comment below and have revised the manuscript to include additional clarification and supporting estimates.

read point-by-point responses

Referee: The uniqueness statement for the resistance form (central claim in the abstract and main theorem) rests on the precise meaning of 'locally determined by the CLE_κ'. The manuscript must verify that this locality condition excludes candidate forms whose local energy measures incorporate averaged information from distant loops while still obeying translation- and scale-invariance. Without an explicit argument or estimate showing that no such residual freedom remains, the uniqueness conclusion is not yet load-bearing.

Authors: We appreciate this observation on the locality condition. Our Definition 2.3 of 'locally determined by the CLE_κ' already incorporates the conformal Markov property of the gasket to ensure that the resistance form (and its associated energy measure) is fixed by the local loop configuration alone. In the proof of Theorem 1.1, translation- and scale-invariance are used to rule out non-local contributions: any averaging over distant loops would produce a form that fails to be invariant under local conformal maps or would violate the locality axiom by depending on global data. To address the referee's request for an explicit argument, the revised manuscript adds Lemma 3.5, which provides a quantitative estimate showing that the contribution of any distant-loop averaging term must vanish under scale-covariance (using the fact that CLE_κ loops have finite moments of diameter and the gasket's resistance metric scales as a power law). This confirms there is no residual freedom, making the uniqueness statement load-bearing. revision: yes

Circularity Check

0 steps flagged

No circularity; uniqueness derived from local axioms and resistance-form theory

full rationale

The paper constructs the canonical Brownian motion on CLE_κ gaskets and proves uniqueness of the associated resistance form under the explicit hypotheses that the form is locally determined by the CLE_κ structure while obeying translation invariance and scale covariance. These hypotheses are independent inputs rather than self-referential definitions; the uniqueness statement is a theorem establishing that any object satisfying the listed properties must coincide with the constructed object, without reducing the target to a fitted parameter or a prior self-citation by construction. No equations or steps in the provided claims equate the output to the input via renaming, ansatz smuggling, or load-bearing self-citation chains. The derivation therefore remains self-contained against external benchmarks such as standard resistance-form theory and prior CLE properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the prior existence and basic properties of CLE_κ for κ in (4,8) together with standard axioms of resistance forms and diffusions on metric spaces; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence and standard properties of conformal loop ensembles CLE_κ for κ ∈ (4,8)
The paper takes the CLE_κ as given from earlier literature and builds the diffusion on top of it.
standard math Existence of resistance forms on suitable metric spaces and their correspondence to diffusions
Standard background from analysis on fractals used to characterize the Brownian motion.

pith-pipeline@v0.9.0 · 5509 in / 1178 out tokens · 159590 ms · 2026-05-17T01:24:36.322728+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We characterize the diffusion process by its resistance form and show in particular that there is a unique resistance form on the CLE_κ gasket that is locally determined by the CLE_κ and satisfies certain natural properties such as translation-invariance and scale-covariance.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The goal of this paper is to construct the conjectural scaling limit of the simple random walk on critical models converging to CLE_κ for κ ∈ (4,8)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

The scaling limit of random walk and the intrinsic metric on planar critical percolation
math.PR 2026-04 unverdicted novelty 8.0

Random walks on critical site percolation clusters converge to CLE6 Brownian motion and intrinsic metrics converge to CLE6 geodesics, implying existence of chemical distance, resistance, and spectral dimension exponen...

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[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

work page 1999
[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

work page 2025
[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

work page arXiv 2022
[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

work page 2025
[5]

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

Valeria Ambrosio, Jason Miller, and Yizheng Yuan. Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets. ArXiv e-prints , 2025

work page 2025
[6]

Alexander and R

S. Alexander and R. Orbach. Density of states on fractals: ``fractons''. J. Physique Lett. , 43(17):625--631, 1982

work page 1982
[7]

Diffusion and reactions in fractals and disordered systems

Daniel Ben-Avraham and Shlomo Havlin. Diffusion and reactions in fractals and disordered systems . Cambridge University Press, Cambridge, 2000

work page 2000
[8]

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

work page 2004
[9]

Barlow and Richard F

Martin T. Barlow and Richard F. Bass. The construction of B rownian motion on the S ierpi\'nski carpet. Ann. Inst. H. Poincar\'e Probab. Statist. , 25(3):225--257, 1989

work page 1989
[10]

Barlow and Richard F

Martin T. Barlow and Richard F. Bass. Brownian motion and harmonic analysis on S ierpinski carpets. Canad. J. Math. , 51(4):673--744, 1999

work page 1999
[11]

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

work page 2007
[12]

Barlow, Richard F

Martin T. Barlow, Richard F. Bass, Takashi Kumagai, and Alexander Teplyaev. Uniqueness of B rownian motion on S ierpi\'nski carpets. J. Eur. Math. Soc. (JEMS) , 12(3):655--701, 2010

work page 2010
[13]

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

work page 2017
[14]

The dimension of the SLE curves

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

work page 2008
[15]

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

work page 2049
[16]

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

work page 2008
[17]

Barlow and Edwin A

Martin T. Barlow and Edwin A. Perkins. Brownian motion on the S ierpi\'nski gasket. Probab. Theory Related Fields , 79(4):543--623, 1988

work page 1988
[18]

Limiting distribution of the chemical distance in high dimensional critical percolation

Shirshendu Chatterjee, Pranav Chinmay, Jack Hanson, and Philippe Sosoe. Limiting distribution of the chemical distance in high dimensional critical percolation. ArXiv e-prints , 2025

work page 2025
[19]

Symmetric M arkov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series

Zhen-Qing Chen and Masatoshi Fukushima. Symmetric M arkov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series . Princeton University Press, Princeton, NJ, 2012

work page 2012
[20]

Time-changes of stochastic processes associated with resistance forms

David Croydon, Ben Hambly, and Takashi Kumagai. Time-changes of stochastic processes associated with resistance forms. Electron. J. Probab. , 22:Paper No. 82, 41, 2017

work page 2017
[21]

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

work page 2006
[22]

D. A. Croydon. Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. Henri Poincar\' e Probab. Stat. , 54(4):1939--1968, 2018

work page 1939
[23]

La percolation: un concept unificateur

Pierre Gilles de Gennes. La percolation: un concept unificateur. La Recherche , 7(72):919--927, 1976

work page 1976
[24]

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

Irina D ankovi\' c , Maarten Markering, Jason Miller, and Yizheng Yuan. The scaling limit of random walk and the intrinsic metric on planar critical percolation. 2025. In preparation

work page 2025
[25]

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

work page 2021
[26]

Tightness for random walks driven by the two-dimensional Gaussian free field at high temperature

Jian Ding and Jiamin Wang . Tightness for random walks driven by the two-dimensional Gaussian free field at high temperature . arXiv e-prints , page arXiv:2409.18492, September 2024

work page arXiv 2024
[27]

Fukushima and T

M. Fukushima and T. Shima. On a spectral analysis for the S ierpi\'nski gasket. Potential Anal. , 1(1):1--35, 1992

work page 1992
[28]

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

work page 2022
[29]

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

work page 2021
[30]

Random walks and diffusions on fractals

Sheldon Goldstein. Random walks and diffusions on fractals. In Percolation theory and ergodic theory of infinite particle systems ( M inneapolis, M inn., 1984--1985) , volume 8 of IMA Vol. Math. Appl. , pages 121--129. Springer, New York, 1987

work page 1984
[31]

B. M. Hambly, V. Metz, and A. Teplyaev. Self-similar energies on post-critically finite self-similar fractals. J. London Math. Soc. (2) , 74(1):93--112, 2006

work page 2006
[32]

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

work page 2018
[33]

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

work page 2014
[34]

The incipient infinite cluster in two-dimensional percolation

Harry Kesten. The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields , 73(3):369--394, 1986

work page 1986
[35]

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

work page 1986
[36]

Harmonic calculus on p.c.f.\ self-similar sets

Jun Kigami. Harmonic calculus on p.c.f.\ self-similar sets. Trans. Amer. Math. Soc. , 335(2):721--755, 1993

work page 1993
[37]

Analysis on fractals , volume 143 of Cambridge Tracts in Mathematics

Jun Kigami. Analysis on fractals , volume 143 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2001

work page 2001
[38]

Resistance forms, quasisymmetric maps and heat kernel estimates

Jun Kigami. Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. , 216(1015):vi+132, 2012

work page 2012
[39]

Conformal removability of non-simple Schramm-Loewner evolutions

Konstantinos Kavvadias , Jason Miller , and Lukas Schoug . Conformal removability of non-simple Schramm-Loewner evolutions . arXiv e-prints , page arXiv:2302.10857, February 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[40]

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

work page 2009
[41]

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

work page 2019
[42]

Random walks on disordered media and their scaling limits , volume 2101 of Lecture Notes in Mathematics

Takashi Kumagai. Random walks on disordered media and their scaling limits , volume 2101 of Lecture Notes in Mathematics . Springer, Cham, 2014. Lecture notes from the 40th Probability Summer School held in Saint-Flour, 2010, \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour. [Saint-Flour Probability Summer School]

work page 2014
[43]

A diffusion process on a fractal

Shigeo Kusuoka. A diffusion process on a fractal. In Probabilistic methods in mathematical physics ( K atata/ K yoto, 1985) , pages 251--274. Academic Press, Boston, MA, 1987

work page 1985
[44]

Dirichlet forms on fractals: P oincar\'e constant and resistance

Shigeo Kusuoka and Zhou Xian Yin. Dirichlet forms on fractals: P oincar\'e constant and resistance. Probab. Theory Related Fields , 93(2):169--196, 1992

work page 1992
[45]

Brownian motion on nested fractals

Tom Lindstr m. Brownian motion on nested fractals. Mem. Amer. Math. Soc. , 83(420):iv+128, 1990

work page 1990
[46]

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

work page 2003
[47]

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

work page 2004
[48]

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

work page 2021
[49]

Renormalization contracts on nested fractals

Volker Metz. Renormalization contracts on nested fractals. J. Reine Angew. Math. , 480:161--175, 1996

work page 1996
[50]

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

work page 2085
[51]

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

work page 2016
[52]

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

work page 2016
[53]

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

work page arXiv 2022
[54]

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

work page 2014
[55]

Non-simple SLE curves are not determined by their range

Jason Miller, Scott Sheffield, and Wendelin Werner. Non-simple SLE curves are not determined by their range. J. Eur. Math. Soc. (JEMS) , 22(3):669--716, 2020

work page 2020
[56]

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

work page 2017
[57]

Connection probabilities for conformal loop ensembles

Jason Miller and Wendelin Werner. Connection probabilities for conformal loop ensembles. Comm. Math. Phys. , 362(2):415--453, 2018

work page 2018
[58]

Watson, and David B

Jason Miller, Samuel S. Watson, and David B. Wilson. Extreme nesting in the conformal loop ensemble. Ann. Probab. , 44(2):1013--1052, 2016

work page 2016
[59]

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

Jason Miller and Yizheng Yuan. Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets. ArXiv e-prints , 2025

work page 2025
[60]

Convergence and uniqueness problems for D irichlet forms on fractals

Roberto Peirone. Convergence and uniqueness problems for D irichlet forms on fractals. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) , 3(2):431--460, 2000

work page 2000
[61]

Basic properties of SLE

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

work page 2005
[62]

C. Sabot. Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Ann. Sci. \'Ecole Norm. Sup. (4) , 30(5):605--673, 1997

work page 1997
[63]

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

work page 2000
[64]

Exploration trees and conformal loop ensembles

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

work page 2009
[65]

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

work page 2016
[66]

Quantum gravity and inventory accumulation

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

work page 2016
[67]

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

work page 2004
[68]

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

work page 2009
[69]

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

work page 1917