Imaginary Gaussian Multiplicative Chaos

Imaginary Gaussian multiplicative chaos is a random generalized function obtained by renormalizing a complex exponential of a Gaussian field. For a centered regularized field and real parameter , its approximants have the form

The positive variance correction makes the expectation equal to 1. For logarithmically correlated fields, the limit is interpreted against test functions or in a space of generalized functions, under the applicable parameter and regularization hypotheses. It is complex valued and differs from the positive random measure of ordinary Gaussian multiplicative chaos.