环形区域中的多个Ising界面与$2N$边径向SLE
Multiple Ising interfaces in annulus and $2N$-sided radial SLE
AI总结：
本文研究了环形区域临界Ising模型在稀有事件下的界面分布，证明了其收敛于$2N$边径向SLE$_3$，并通过对多重弦SLE的估计推广了格林函数的结论。
摘要：
我们考虑环形区域中的临界平面Ising模型，其外边界具有交替边界条件，内边界具有自由边界条件。随着内孔尺寸趋于零，所有界面在相互交汇前均靠近内孔的事件是一个稀有事件。我们证明了，在此稀有事件条件下，界面集合的分布在全变差距离下收敛于由[HL21]引入的所谓$2N$边径向SLE$_3$。该证明关键依赖于对多重弦SLE的估计。假设$(γ_1, \ldots, γ_N)$是单位圆盘中的弦$N$-SLE$_κ$，其中$κ\in (0,4]$，我们考虑所有$N$条曲线均靠近原点的概率。我们证明了极限$\lim_{r\to 0+}r^{-A_{2N}}\mathbb{P}[\mathrm{dist}(0,γ_j)<r, 1\le j\le N]$存在，其中$A_{2N}$是所谓的$2N$臂指数，$\mathrm{dist}$是欧几里得距离。我们将该极限称为弦$N$-SLE$_κ$的格林函数。该估计推广了先前分别在[LR12, LR15]和[Zha20]中证明的$N=1$和$N=2$的结论。
英文摘要：
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 $(γ_1, \ldots, γ_N)$ is chordal $N$-SLE$_κ$ with $κ\in (0,4]$ in the unit disc, and we consider the probability that all $N$ curves get close to the origin. We prove that the limit $\lim_{r\to 0+}r^{-A_{2N}}\mathbb{P}[\mathrm{dist}(0,γ_j)<r, 1\le j\le N]$ 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$_κ$. This estimate is a generalization of previous conclusions with $N=1$ and $N=2$ proved in~[LR12, LR15] and~[Zha20] respectively.
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