The critical Ising magnetization field can be reconstructed from its $+/-$ interfaces

Abstract: The critical Ising model admits several natural scaling limits. For an Ising model on a lattice approximation $D<sup>a$ 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.