Convergence of two-dimensional lattice-model interfaces to Schramm–Loewner evolution

Establish convergence of the relevant interfaces in two-dimensional lattice models to Schramm–Loewner Evolution (SLE), beyond the limited collection of models for which such convergence is currently known.

Background

The paper situates planar percolation within the broader study of two-dimensional lattice models and their anticipated conformal scaling limits. It identifies convergence of interfaces to SLE as a wide-open conjectural phenomenon and notes that rigorous proofs exist only for a small number of models, including critical Ising interfaces on isoradial graphs and critical site percolation on the triangular lattice.

References

These emergent symmetries have lead to the wide-open conjecture that certain interfaces converge to the Schramm--Loenwer Evolution (SLE). This has been proved in only a handful of models, including critical Ising model on isoradial graphs and critical site percolation on the triangular lattice.

— Planar percolation and the loop O(n) model  (2508.20917 - Glazman et al., 28 Aug 2025) in Section 1, Introduction

Although such a discrete realization remains open, Proposition~\ref{prop::GFF_rad} provides a continuum counterpart at $\kappa=4$: when $\sigma_1=\cdots=\sigma_n=1$, the function $F{n}{(\sigma; \mu)}$ in~eqn::LFrad_sigma_def coincides with the partition function of multi-radial $\mathrm{SLE}_4$ with spiral i.e. the partition function $Z{n}{(\mu)}$ in~eqn::LZrad_mu_U with $\kappa=4$, and the corresponding SLE dynamics arise naturally as level lines of the Gaussian free field.

eqn::LZrad_mu_U:

Zn(μ)(θ):=∏1≤i<ℓ≤n∣eiθℓ−eiθi∣2κ×exp⁡(μκ∑j=1nθj),for θ=(θ1,…,θn)∈XnU.Z{n}^{(\mu)}({\theta}) := \prod_{1\le i<\ell\le n}|e^{i\theta_{\ell}}-e^{i\theta_i}|^{\frac{2}{\kappa}} \times \exp\left( \frac{\mu}{\kappa}\sum_{j=1}^{n} \theta_j \right), \qquad \text{for }{\theta}=(\theta_1, \ldots, \theta_n) \in X_n^{U}.

— Classification of commutation relation for multi-radial SLE  (2609.19739 - Huang et al., 17 Sep 2026) in Section 1, subsection “GFF level lines”

SLEs and CLEs are conjectured (and in several cases rigorously shown) to describe the scaling limits of interfaces in critical statistical mechanics models on planar lattices as well as on random planar maps.

— The conformally invariant metric on CLE$_4$ III: uniqueness  (2609.04140 - Kammerer et al., 3 Sep 2026) in Section 1, Introduction, subsection “Overview”

SLEs and CLEs are conjectured (and in several cases rigorously shown) to describe the scaling limits of interfaces in critical statistical mechanics models on planar lattices as well as on random planar maps.

— The conformally invariant metric on CLE$_4$ II: existence of geodesics  (2609.04139 - Kammerer et al., 3 Sep 2026) in Section 1, Introduction, Overview