Abstract
We consider finite range Gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi-invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs fields; in this context the tree decay of semi-invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so-called Griffiths' phase when analyticity arguments fail.
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Bertini, L., Cirillo, E.N.M. & Olivieri, E. A Combinatorial Proof of Tree Decay of Semi-Invariants. Journal of Statistical Physics 115, 395–413 (2004). https://doi.org/10.1023/B:JOSS.0000019813.58778.bf
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DOI: https://doi.org/10.1023/B:JOSS.0000019813.58778.bf