Let us suppose a complex random function $\displaystyle \Psi (\mathbf{x})$ and its complex conjugate $\displaystyle \Psi ^{*}(\mathbf{x})$. We assume that these functions possess gaussian statistics, so that mean value of some functional $\displaystyle f\left[ \Psi ,\Psi ^{*}\right]$ shall be calculated as \begin{equation*} \langle f\rangle =\dfrac{\int D\Psi ^{*} D\Psi \ f\left[ \Psi ,\Psi ^{*}\right] \ \exp\left[ -\int \mathrm{d}^{n}\mathbf{x} \ \Psi ^{*}\hat{K} \Psi \right]}{\int D\Psi ^{*} D\Psi \ \exp\left[ -\int \mathrm{d}^{n}\mathbf{x} \ \Psi ^{*}\hat{K} \Psi \right]} \end{equation*} Here $\hat{K}$ is some differential operator.
The question: if I know for certain, that for the function $\displaystyle \Psi (\mathbf{x})$ there is such a point $\displaystyle \mathbf{y}$, that $\displaystyle \Psi (\mathbf{y}) =0$, how can I calculate a probability distibustion function for $\displaystyle \mathbf{y}$? Shall it be
\begin{equation*} P(\mathbf{y}) =\langle \delta ( \Re \Psi (\mathbf{y})) \cdotp \delta ( \Im \Psi (\mathbf{y})) \rangle \end{equation*} or something else?
Thanks everybody in advance!