Abstract
The high-temperature expansion is used, to all orders, to obtain an equivalent description of the $X\ensuremath{-}Y$ model in terms of a quantized vector field interacting with itself and with a system of quantized charges. This description is used to all orders to obtain some exact and some approximate results. The exact results are (a) an alternative description in terms of integer Ising spins interacting via a nearest-neighbor interaction; (b) upper bounds on the correlation length and transition temperature in two and three dimenisions. It is also shown that at low temperatures the system can be viewed as a system composed of a few kinds of charges (the number depends on the dimensionality), where the interaction between charges of a given kind is electrostatic, while the interaction between charges of different kinds is zero.
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