The sharp factor-of-IID threshold for the free Ising model on regular trees
For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when \(\tanh\beta\le(d-1)^{-1/2}\). We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.
Cite
@misc{OAI:The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026,
author = {{OpenAI}},
title = {{The sharp factor-of-IID threshold for the free Ising model on regular trees}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf}{OAI:The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026}},
year = {2026}
}