TY - JOUR

T1 - Ising Model on Trees and Factors of IID

AU - Nam, Danny

AU - Sly, Allan

AU - Zhang, Lingfu

N1 - Funding Information:
AS would like to thank Russ Lyons and Yuval Peres for many discussions about the problem. DN is supported by a Samsung Scholarship. AS is supported by NSF Grants DMS-1352013 and DMS-1855527, Simons Investigator Grant and a MacArthur Fellowship. The authors also thank anonymous referees for reading this paper carefully and providing valuable comments.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/1

Y1 - 2022/1

N2 - We study the ferromagnetic Ising model on the infinite d-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature β≥ 0 satisfies tanh β≤ (d- 1) - 1. However, in the reconstruction regime (tanhβ>(d-1)-12), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons (Comb Probab Comput 2(2):285–300, 2017). The solution { Xt(v) } of the SDE is distributed as Xt(v)=tτv+Bt(v),where { τv} is an Ising sample and { Bt(v) } are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever tanhβ≤c(d-1)-12, where c> 0 is an absolute constant.

AB - We study the ferromagnetic Ising model on the infinite d-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature β≥ 0 satisfies tanh β≤ (d- 1) - 1. However, in the reconstruction regime (tanhβ>(d-1)-12), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons (Comb Probab Comput 2(2):285–300, 2017). The solution { Xt(v) } of the SDE is distributed as Xt(v)=tτv+Bt(v),where { τv} is an Ising sample and { Bt(v) } are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever tanhβ≤c(d-1)-12, where c> 0 is an absolute constant.

UR - http://www.scopus.com/inward/record.url?scp=85123080486&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85123080486&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04260-2

DO - 10.1007/s00220-021-04260-2

M3 - Article

AN - SCOPUS:85123080486

VL - 389

SP - 1009

EP - 1046

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -