ProofAtlas

Probability, lattice models, and conformal invariance

Planar Self-Avoiding-Walk Scaling Limit

Collaboration beta

Do long critical self-avoiding lattice paths converge to the conformally invariant SLE₈/₃ random curve?

γδ⇒SLE8/3(D;a,b)
Known results and sources
A critical honeycomb self-avoiding path crosses a simply connected Jordan domain between distinct marked boundary prime ends toward an SLE-like curve silhouette, with convergence marked open.
Between distinct marked boundary prime ends of a simply connected Jordan domain, the conjecture predicts convergence of the critical trace to chordal SLE₈/₃.

Research problem

Exact mathematical statement

Let D⊂ℂD\subset\mathbb C be a simply connected Jordan domain with marked boundary prime ends a,ba,b, and let DδD_\delta be a honeycomb-lattice approximation. At

xc=12+2,x_c=\frac1{\sqrt{2+\sqrt2}},

give each self-avoiding walk γ:aδ→bδ\gamma:a_\delta\to b_\delta probability proportional to xc|γ|x_c^{|\gamma|}. The trace conjecture is

γδ⇒SLE8/3(D;a,b)\gamma_\delta\Longrightarrow\operatorname{SLE}_{8/3}(D;a,b)

in the uniform topology on curves modulo increasing reparametrization, or an equivalent unparametrized-curve topology.

Problem infographic

Problem at a glance

A problem-first explainer fixes a simply connected Jordan domain D with distinct marked boundary prime ends a and b, then shows the exact critical honeycomb law, mesh refinement, and the open unparametrized SLE₈/₃ trace limit.
In a simply connected Jordan domain with distinct marked boundary prime ends, critical honeycomb paths are weighted by the exact fugacity and conjectured to converge as traces to chordal SLE₈/₃.

Current mathematical picture

Where work on Planar Self-Avoiding-Walk Scaling Limit stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Naively switching the tails of two self-avoiding paths at their first intersection automatically preserves self-avoidance and weight. Obtain uniform lattice compactness and a nondegenerate projective first-variation estimate in evolving slit domains, then justify their continuum and local-to-global passages.

Route status · Narrowed route
Main reductionCurrent reduction

Prove Loewner compactness, positive projective four-point covariance, a regular endpoint cocycle, stopped local-L1 martingale convergence, domain-uniformity, exact-restriction passage by a hull sandwich, and local-to-global projective integration.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Planar Self-Avoiding-Walk Scaling Limit in numbers

909retained lines of mathematical investigation909 in the current working snapshot
Argument development
721 · 79%
Explored or eliminated routes
12 · 1%
Computational analysis
3 · 0%
Open obligations
21 · 2%
Definitions and setup
152 · 17%
7selected mapped statements1routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGregory F. Lawler, Oded Schramm, and Wendelin WernerProposed2002Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Planar Self-Avoiding-Walk Scaling LimitA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do critical planar self-avoiding walks converge to SLE₈/₃? — Depends on missing premiseDo critical planarself-avoiding walks convergeto…Conditional SLE identification — Depends on missing premiseConditional SLEidentificationCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetCritical trace conjecture — Depends on missing premiseCritical trace conjectureExact lattice restriction — Depends on missing premiseExact lattice restrictionLimit and switching corrections — Depends on missing premiseLimit and switchingcorrectionsSource-reported limitation — stoppedSource-reported limitationProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit. — OpenProve tightness andcontinuous chordal Loewnergeneration…Prove projective four-point covariance with one common nondegenerate positive exponent and a valid local-to-global domain-deformation theorem. — OpenProve projective four-pointcovariance with one commonnondegenerate…Pass target-change martingales and exact restriction to the continuum with endpoint and hull-limit controls. — OpenPass target-changemartingales and exactrestriction…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Naively switching the tails of two self-avoiding paths at their first intersection automatically preserves self-avoidance and weight. Obtain uniform lattice compactness and a nondegenerate projective first-variation estimate in evolving slit domains, then justify their continuum and local-to-global passages.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.Suggested move: State and verify a multi-crossing compactness criterion uniform over the discrete domain-Markov slit domains used by target-change exploration.
Ready to work on
02
Prove projective four-point covariance with one common nondegenerate positive exponent and a valid local-to-global domain-deformation theorem.Suggested move: Establish the mixed projective Ward variation in the mesh-before-small-hull limit and integrate it with summable uniformization errors.
Ready to work on
03
Pass target-change martingales and exact restriction to the continuum with endpoint and hull-limit controls.Suggested move: Use stopped local-L1 convergence and inner/outer hull sandwiches rather than treating avoidance as an ordinary continuity event.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The exact honeycomb critical constant is proved, but fixed-lattice planar self-avoiding-walk trace convergence to SLE8/3 remains conjectural.

[1][2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedThe exact honeycomb connective constant was proved; the paper explicitly notes that completing the missing discrete-holomorphic relation would also imply SLE8/3 convergence, so the trace limit is not proved…[2]
  2. PreprintLawler, Schramm, and Werner formulated planar SAW scaling-limit conjectures in terms of SLE and conformal restriction.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusPlanar Self-Avoiding-Walk Scaling-Limit Conjecture
Dependency or reductionHoneycomb connective constant

The exact connective constant identifies the critical honeycomb fugacity used by the scaling-limit statement.

[2]
Related problemSelf-avoiding polygon and radial variants

The corresponding self-avoiding polygon and whole-plane/radial variants are related but are not the same chordal fixed-domain statement.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetNo statement-aligned formalization of fixed-lattice SAW trace convergence to SLE8/3 was located.
  • Formalization targetFormal probability measures on unparametrized curves, tightness, Loewner evolution, and critical lattice estimates would be required.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma3 of 73
  • negative result1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementDo critical planar self-avoiding walks converge to SLE₈/₃?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementCritical trace conjectureintermediate
  • retained route statementExact lattice restrictionintermediate
  • retained route statementConditional SLE identificationintermediate
  • retained route statementLimit and switching correctionsintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureSource-reported limitationreported failure
  • Research targetProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.open
  • Research targetProve projective four-point covariance with one common nondegenerate positive exponent and a valid local-to-global domain-deformation theorem.open
  • Research targetPass target-change martingales and exact restriction to the continuum with endpoint and hull-limit controls.open
  • Research targetLoewner compactnesssuperseded
  • Research targetProjective covariance and passagesuperseded
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Naively switching the tails of two self-avoiding paths at their first intersection automatically preserves self-avoidance and weight. Obtain uniform lattice compactness and a nondegenerate projective first-variation estimate in evolving slit domains, then justify their continuum and local-to-global passages.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.

Planar Self-Avoiding-Walk Scaling Limit · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Do long critical self-avoiding lattice paths converge to the conformally invariant SLE₈/₃ random curve?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On the scaling limit of planar self-avoiding walkoriginal source · Gregory F. Lawler, Oded Schramm, Wendelin Werner · arXiv · 2002 · ARXIV math/0204277 · accessed Aug 14, 2026
  2. 2
    The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2))peer reviewed result · Hugo Duminil-Copin, Stanislav Smirnov · Annals of Mathematics · 2012-05-01 · ARXIV 1007.0575 · DOI 10.4007/annals.2012.175.3.14 · MR 2912714 · accessed Aug 14, 2026

Important qualifications

  • Scoped to fixed planar-lattice trace convergence and the exact honeycomb critical constant.
  • Random planar-map SLE results and numerical studies were not treated as proofs or solved special cases of the fixed-lattice statement.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image