Multiple Ising interfaces in annulus and $2N$-sided radial SLE

TL;DR

This paper proves that the collection of multiple Ising interfaces in an annulus, conditioned on a rare event, converges to a specific radial SLE process, and establishes a Green's function estimate for multiple chordal SLEs.

Contribution

It introduces a convergence result for multiple Ising interfaces to $2N$-sided radial SLE and generalizes Green's function estimates for multiple chordal SLEs.

Findings

Convergence of Ising interfaces to $2N$-sided radial SLE$_3$.

Existence of the Green's function limit for multiple chordal SLE.

Generalization of previous Green's function estimates for $N=1,2$.

Abstract

We consider critical planar Ising model in annulus with alternating boundary conditions on the outer boundary and free boundary conditions in the inner boundary. As the size of the inner hole goes to zero, the event that all interfaces get close to the inner hole before they meet each other is a rare event. We prove that the law of the collection of the interfaces conditional on this rare event converges in total variation distance to the so-called $2N$-sided radial SLE$_3$, introduced by~[HL21]. The proof relies crucially on an estimate for multiple chordal SLE. Suppose $(\gamma_1, \ldots, \gamma_N)$ is chordal $N$-SLE$_{\kappa}$ with $\kappa\in (0,4]$ in the unit disc, and we consider the probability that all $N$ curves get close to the origin. We prove that the limit $\lim_{r\to 0+}r^{-A_{2N}}\mathbb{P}[\mathrm{dist}(0,\gamma_j)<r, 1\le j\le N]$ exists, where $A_{2N}$ is the so-called $2N$-arm exponents and $\mathrm{dist}$ is Euclidean distance. We call the limit Green's function for chordal $N$-SLE$_{\kappa}$. This estimate is a generalization of previous conclusions with $N=1$ and $N=2$ proved in~[LR12, LR15] and~[Zha20] respectively.