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The Lee-Yang property of certain moment generating functions having only pureimaginary zeros is valid for Ising type models with one-component spins and XYmodels with two-component spins. Villain models and complex Gaussianmultiplicative chaos are two-component systems analogous to XY models andrelated to Gaussian free fields. Although the Lee-Yang property is known to bevalid generally in the first case, we show that is not so in the second. Ourproof is based on two theorems of general interest relating the Lee-Yangproperty to distribution tail behavior.

Lee-Yang Property and Gaussian multiplicative chaos