[Paper Review] On the Uniqueness of Global Multiple SLEs

Vincent Beffara, Eveliina Peltola+1 more|arXiv (Cornell University)|Jan 23, 2018

Stochastic processes and statistical mechanics41 references12 citations

TL;DR

This paper establishes the uniqueness of global multiple Schramm-Loewner Evolutions (SLE) for κ ∈ (0, 4] by proving that a probability measure on collections of pairwise disjoint curves with a specific conditional law property is uniquely determined by its connectivity pattern. The proof uses a Markov chain argument for N=2 and generalizes it to N≥3, yielding exponential convergence rates. The result implies the convergence of multiple interfaces in critical Ising, FK-Ising, and percolation models to global multiple SLEκ with κ=3 and κ=16/3 respectively.

ABSTRACT

This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising, FK-Ising, and percolation models.

Motivation & Objective

• To establish a minimal set of characterizing properties for global multiple SLEs that uniquely determine the law of collections of non-intersecting curves.
• To prove the uniqueness of global multiple SLEκ for κ ∈ (0, 4] with a given link pattern α ∈ LPN, extending beyond existing constructions.
• To apply the uniqueness result to prove convergence of multiple interfaces in critical planar lattice models such as the Ising and FK-Ising models to global multiple SLEκ.
• To extend the classification to κ ∈ (4, 6] using discrete model information and RSW-type estimates, conditional on single-interface convergence.
• To identify the marginal law of a single curve in the scaling limit as a weighted chordal SLE3 in the Ising model.

Proposed method

• Define global N-SLEκ as a probability measure on families of pairwise disjoint curves connecting boundary points in a given link pattern α ∈ LPN, with the conditional law of each curve being chordal SLEκ in the remaining domain.
• Prove uniqueness via a Markov chain argument for N=2, showing exponential convergence to the invariant measure corresponding to the global multiple SLE.
• Generalize the uniqueness proof to arbitrary N ≥ 3 using conditional independence and coupling techniques based on SLE boundary perturbation.
• Use the convergence of a single critical Ising interface to chordal SLE3 (from CDCH+14) as a foundation for proving joint convergence of multiple interfaces.
• Apply RSW-type estimates and domain Markov properties in discrete models (Ising and FK-Ising) to control interface behavior and rule out pathological limits.
• Leverage conformal invariance and the Hamiltonian structure of the Ising model to relate discrete interfaces to SLE via the Edwards-Sokal coupling.

Experimental results

Research questions

• RQ1 What minimal set of properties uniquely characterizes global multiple SLEs for κ ∈ (0, 4]?
• RQ2 Can the uniqueness of global multiple SLEs be established for N ≥ 3 using methods beyond the Gaussian free field coupling used for N=2?
• RQ3 Does the convergence of a single interface to SLEκ in critical lattice models imply the convergence of multiple interfaces to global multiple SLEκ?
• RQ4 How can the classification of global multiple SLEs be extended to κ ∈ (4, 6], where explicit constructions fail?
• RQ5 What is the marginal law of a single curve in the scaling limit of multiple interfaces in the critical Ising model?

Key findings

• For any κ ∈ (0, 4] and any link pattern α ∈ LPN, there exists a unique global N-SLEκ measure on families of pairwise disjoint curves with the specified conditional law property.
• The uniqueness proof for N ≥ 3 is established via a generalized Markov chain argument, with an exponential convergence rate to the invariant measure.
• Multiple interfaces in the critical Ising model with alternating boundary conditions converge weakly to global multiple SLE3 as δ → 0, conditionally on a fixed link pattern.
• The marginal law of a single curve in the global multiple SLE3 is identified as a weighted chordal SLE3, consistent with known single-interface results.
• For κ ∈ (4, 6], the existence and uniqueness of global multiple SLEκ are established conditionally on the convergence of a single interface and RSW estimates, with the FK-Ising model (q=2) yielding κ=16/3.
• The convergence of multiple interfaces in critical percolation (κ=6) follows from the same framework, as shown by the convergence of a single interface to SLE6.