[Paper Review] Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$

Kôhei Uchiyama|arXiv (Cornell University)|May 9, 2017

Stochastic processes and statistical mechanics11 references3 citations

TL;DR

This paper establishes the convergence of loop-erased random walk (LERW) on planar graphs to chordal SLE₂ under general conditions, by proving that conditioned random walk excursions leave the initial boundary neighborhood through the intrinsic interior with high probability. The key contribution is a harmonic measure estimate that removes the analyticity assumption required in prior work, enabling uniform convergence to chordal SLE₂ in simply connected grid domains without restrictive boundary smoothness.

ABSTRACT

This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected grid domain and conditioned to exit it through another boundary vertex specified in advance. The primary contribution of the present paper is an estimate, which states that the excursion of the conditioned walk leaves an intrinsic neighborhood of its initial point not 'along' the boundary but through an intrinsic interior of the domain with high probability. Based on this result we give a proof for the convergence to the chordal SLE, a result that has recently been proved by Suzuki [12] under an analyticity assumption on the boundary of the domain arising in the limit.

Motivation & Objective

• To establish the convergence of loop-erased random walk (LERW) on planar graphs to chordal SLE₂ under minimal assumptions.
• To remove the analyticity assumption on the boundary of the domain required in prior results, such as those by Suzuki (2016).
• To provide a uniform estimate on the harmonic measure of random walk excursions in grid domains, showing they exit the initial boundary neighborhood through the intrinsic interior with high probability.
• To extend the convergence result from radial to chordal SLE₂ in the context of general random walks on planar graphs satisfying the invariance principle.
• To verify weak convergence of the loop-erased path to chordal SLE₂ using the driving function convergence and time-reversal symmetry of LERW.

Proposed method

• Establishes a harmonic measure estimate (Proposition 4.6) showing that a conditioned random walk starting near a boundary vertex of a simply connected grid domain exits an intrinsic neighborhood of its starting point through the domain’s interior with high probability.
• Uses the invariance principle (Hypothesis H) to ensure weak convergence of scaled random walk paths to Brownian motion, enabling the use of conformal invariance in the scaling limit.
• Applies conformal mapping techniques via the Riemann map φ: D → 𝔻 to transfer the problem to the unit disk, where chordal SLE₂ is well-defined from φ(a) to φ(b) for prime ends a, b ∈ ∂ₚₙ𝒹D.
• Leverages time-reversal symmetry of LERW under symmetric transition probabilities (p(u,v) = p(v,u)) to relate the law of the loop-erased path to that of the time-reversed excursion.
• Applies Shelefild and Sun’s result (Corollary 1.7) to deduce weak convergence of the loop-erased path under the metric d*ₚ, given convergence of the driving function.
• Uses Carathéodory convergence of domains Dₙ/ρₙ → D to ensure the conformal maps φₙ converge uniformly on compact subsets, enabling the limit to be identified as chordal SLE₂.

Experimental results

Research questions

• RQ1 Under what conditions does loop-erased random walk on a planar graph converge to chordal SLE₂?
• RQ2 Can the convergence of LERW to chordal SLE₂ be established without assuming analyticity of the domain boundary?
• RQ3 How does the harmonic measure of a conditioned random walk excursion behave near the boundary of a simply connected grid domain?
• RQ4 What is the probability that a random walk excursion starting near the boundary leaves the initial intrinsic neighborhood through the interior rather than along the boundary?
• RQ5 Can the convergence of the driving function of the LERW be established uniformly across large grid domains with boundary conditions?

Key findings

• The primary contribution is a sharp harmonic measure estimate (Proposition 4.6) showing that a conditioned random walk excursion from a boundary vertex leaves an intrinsic neighborhood of its starting point through the domain’s interior with high probability.
• The paper removes the analyticity assumption on the boundary of the domain required in Suzuki’s (2016) result, thereby generalizing the convergence of LERW to chordal SLE₂ to a broader class of domains.
• The convergence of the loop-erased path to chordal SLE₂ is established in law under the metric d*ₚ on the space of continuous curves in the unit disk.
• The driving function of the LERW converges weakly to that of chordal SLE₂, which implies convergence of the Loewner chains.
• The result holds under the invariance principle (Hypothesis H) and symmetry of transition probabilities, with convergence uniform in the domain size as ρₙ → ∞.
• The proof relies on the time-reversal symmetry of LERW and the fact that the law of LE(Γ⁻) under the conditional measure coincides with that of LE(Γ) under the time-reversed measure, enabling the use of duality arguments.