发表机构
Université Claude Bernard Lyon 1; CNRS UMR 5208, Institut Camille Jordan; Courant Institute (NYU); Universidad de Chile, Centro de Modelamiento Matemático (AFB170001), UMI-CNRS 2807（里昂第一大学; 法国国家科学研究中心联合研究实验室5208，卡米尔·若尔当研究所; 纽约大学库朗数学科学研究所; 智利大学，数学建模中心（AFB170001），中法联合研究国际单位2807）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明临界Ising磁化场可从嵌套CLE_3界面直接重构，补充了从CLE_{16/3}构造的结果，并猜想其逆命题成立。
AI 中文摘要
临界Ising模型具有多种自然标度极限。对于平面域$D$的格点近似$D^a$上的Ising模型，当$a\ o0$时，重标度自旋场收敛到临界Ising磁化场（IMF），即$D$上的一个粗糙随机分布，而自旋界面收敛到嵌套的$CLE_3$，FK-Ising表示收敛到彩色的$CLE_{16/3}$。本文证明IMF可直接从嵌套的$CLE_3$重构。这补充了先前从彩色$CLE_{16/3}$构造IMF的结果[CGN15]。我们的结果还表明，若通过CLE渗流[MSW17]从嵌套的$CLE_3$构造彩色$CLE_{16/3}$，则所得构造恢复相同的磁化场。本重构的主要困难在于IMF的重整化指数$15/8$小于$CLE_3$地毯的Hausdorff维数$187/96$。因此，不能像$CLE_{16/3}$那样通过直接的Minkowski含量构造从嵌套$CLE_3$恢复该场。我们猜想其逆可测性也成立，即嵌套$CLE_3$本身可从IMF重构。
英文摘要
The critical Ising model admits several natural scaling limits. For an Ising model on a lattice approximation $D^a$ of a planar domain $D$, the rescaled spin field converges, as $a\to0$, to the critical Ising magnetization field (IMF), a rough random distribution on $D$, while the spin interfaces converge to a nested $CLE_3$ and the FK--Ising representation converges to a colored $CLE_{16/3}$. In this work, we prove that the IMF can be directly reconstructed from the nested $CLE_3$. This complements the previously known construction of the IMF from the colored $CLE_{16/3}$ [CGN15]. Our result also implies that %We further show that if one constructs a colored $CLE_{16/3}$ from the nested $CLE_3$ through CLE percolation [MSW17], the resulting construction recovers the same magnetization field. A main difficulty in this present reconstruction comes from the fact that the renormalization exponent of the IMF, $15/8$, is smaller than the Hausdorff dimension of the $CLE_3$ carpet, $187/96$. Consequently, the field cannot be recovered by a direct Minkowski-content construction from the nested $CLE_3$ as is the case for $CLE_{16/3}$. We conjecture that the converse measurability also holds, namely that the nested $CLE_3$ can itself be reconstructed from the IMF.
Comments49 pages, 5 figures. The main results of this paper were announced in June 2025
