ABSENCE OF PHASE TRANSITIONS ON TREE STRUCTURES

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Abstract
We rigorously prove that the correlation functions of any statistical model having a compact transitive symmetry group and nearest-neighbor interactions on any tree structure are equal to the corresponding ones on a linear chain. The exponential decay of the latter implies the absence of long-range order on any tree. On the other hand, for trees with exponential growth such as Bethe lattices, one can show the existence of a particular kind of mean field phase transition without long-range order.

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