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what is Lee-Yang circle theorem and what is it used for ?? , i mean given a measure how can you know that is Ferromagnetic and hence all its zeros lie on a Circle ??

the Lee-Yang circle theorem proof is it only valid for the measure $ \delta (x-1) + \delta (x+1) $

how can you proof the Circle theorem for any measure ?? if any measure is positive and EVEN does Lee-Yang circle theorem hold ??

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You can start here to get an overview.

From a high level the L-Y Circle theorem is a statement about the location of the zeros of the partition function used in statistical field theory for ferromagnetic systems. It applies when using a mean-field approximation and the zeros of the partition function end up constrained to the unit circle. I can't remember the details of how the proof goes but Itzykson and Drouffe has a very detailed section on the Lee-Yang zeros. Section 3.2 (if I remember correctly) will answer most of your detailed questions as it has a full on proof of the theorem.

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L-Y theorems applies much more generally than for mean-fields. As an observation, it is obviously useful for basically any problem but even rigorously it has been extended to much much more (e.g. finite-size corrections have also been computed). –  Marek Aug 21 '11 at 8:45
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