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 Da of a planar domain D, the rescaled spin field converges, as a→0, to the critical Ising magnetization field (IMF), a rough random distribution on D, while the spin interfaces converge to a nested CLE3 and the FK--Ising representation converges to a colored CLE16/3. In this work, we prove that the IMF can be directly reconstructed from the nested CLE3. This complements the previously known construction of the IMF from the colored CLE16/3 [CGN15]. Our result also implies that %We further show that if one constructs a colored CLE16/3 from the nested CLE3 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 CLE3 carpet, 187/96. Consequently, the field cannot be recovered by a direct Minkowski-content construction from the nested CLE3 as is the case for CLE16/3. We conjecture that the converse measurability also holds, namely that the nested CLE3 can itself be reconstructed from the IMF.
Related
- Near-critical Ising interfaces in the thermal direction: Absolute continuity with respect to SLE3
- Analytical results for the Shannon entropy of the critical transverse-field Ising chain
- Critical Ising model as the M(4,3) minimal model of conformal field theory
- Uniform Bessel bounds for magnetization crossings in finite Ising chains
- Normal approximation for total magnetization in subcritical long-range Ising models
BibTeX
@misc{cahen2026critical,
title = {{The critical Ising magnetization field can be reconstructed from its $+/-$ interfaces}},
author = {Paul Cahen and Christophe Garban and Avelio Sepúlveda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02064},
eprint = {2610.02064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02064},
}
RIS
TY - UNPB
TI - The critical Ising magnetization field can be reconstructed from its +/− interfaces
AU - Paul Cahen
AU - Christophe Garban
AU - Avelio Sepúlveda
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02064
UR - https://arxiv.org/abs/2610.02064
AB - 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→0, 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.
ER -