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For the nearest neighbours Ising-Model in any dimension, it is known that $$ \mu^{free}_\beta= \frac{1}{2} \mu^{+}_\beta+\frac{1}{2} \mu^{-}_\beta $$ for any inverse of temperature $\beta>0$.

Is the same valid for dimension $d=1$, with a long-range interaction $J(i,j)=\frac{1}{|i-j|^\alpha}1_{i\neq j} $ for $\alpha \in (1,2]$? I remember a professor once told me the Ising Model with long-range interaction in dimension $d$ roughly behaves like the nearest neighbours Ising Model in dimension $d+1$. So I supposed we would expect such phenomena but are there strong reasons to believe so? Is it proved? At least maybe for a range of temperatures? I appreciate any references.

Kernel
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  • What exactly is $\mu_\beta$ here? – By Symmetry Jan 31 '19 at 15:41
  • By $\mu^#\beta$ I am denoting the weak limit of the measures $\mu^#{\Lambda_n,\beta}$ for some $\Lambda_n \uparrow \mathbb{Z}$. Where $#$ is the boundary condition. $# = +,-$ means setting all the spins out of $\Lambda_n$ as $+1$ or $-1$. And for $#=free$ I mean to disregard the interactions between the pairs of spins with one side inside and another outside $\Lambda_n$. – Kernel Jan 31 '19 at 16:03

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Yes, this follows from a recent result by Aran Raoufi. His result is much more general and applies to an extremely large class of graphs.

Yvan Velenik
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