{"id":"openai-math/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026","tags":["probability","statistical-mechanics","ising-model"],"score":0,"title":"The sharp factor-of-IID threshold for the free Ising model on regular trees","domain":"Probability and statistical mechanics","family":"236","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":true,"family_title":"The exact factor-of-IID threshold for free Ising spins on trees"}