Supercritical sharpness for Ising double random currents
Abstract
We show that the percolation model induced by the double random current representation of the Ising model on ℤd, with d≥ 3, satisfies local uniqueness of the macroscopic cluster with high probability for every β>βc, uniformly in boundary conditions (i.e. sources). Such a result is sometimes called supercritical sharpness in percolation theory and yields a detailed description of the model via standard renormalisation techniques. As applications, we show the exponential decay of truncated correlations for the Ising model and exponential mixing for the FK-Ising measure, results previously obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is considerably simpler, as it replaces the use of a highly technical multi-valued map principle with a simple sprinkling argument.
Related
- Double cluster swapping for spin models: Pfaffian relations and sharpness
- Double Yangian and reflection algebras of the Lie superalgebra glM|N, II: Quantum currents
- Supercritical Branching Random Walks on Nested Lattices
- Critical oriented percolation enhanced along random supercritical horizontal lines
- Left tail asymptotics for the density of supercritical branching processes in random environment
BibTeX
@misc{gunaratnam2026supercritical,
title = {{Supercritical sharpness for Ising double random currents}},
author = {Trishen S. Gunaratnam and Christoforos Panagiotis and Romain Panis and Franco Severo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20583},
eprint = {2610.20583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20583},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Supercritical sharpness for Ising double random currents
AU - Trishen S. Gunaratnam
AU - Christoforos Panagiotis
AU - Romain Panis
AU - Franco Severo
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20583
UR - https://arxiv.org/abs/2610.20583
AB - We show that the percolation model induced by the double random current representation of the Ising model on ℤ^d, with d≥ 3, satisfies local uniqueness of the macroscopic cluster with high probability for every β>β_c, uniformly in boundary conditions (i.e. sources). Such a result is sometimes called supercritical sharpness in percolation theory and yields a detailed description of the model via standard renormalisation techniques. As applications, we show the exponential decay of truncated correlations for the Ising model and exponential mixing for the FK-Ising measure, results previously obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is considerably simpler, as it replaces the use of a highly technical multi-valued map principle with a simple sprinkling argument.
ER -