I'm reading an article about Non-Gaussianity of Large-Scale Cosmic Microwave Background (link) and the authors write that the n-point correlation function of $e^{\varphi(x)}$ where $\varphi(x)$ is a random Gaussian field can be written as \begin{equation} \langle e^{\varphi(x_1)}\ldots e^{\varphi(x_N)}\rangle=e^{\frac{1}{2}\int dx dyJ(x)\langle\varphi(x)\varphi(y)\rangle J(y)} \end{equation} where $J(x)=\sum_i^N \delta(x-x_i)$. I can't understand how to get this relation and what is its physical meaning.
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It's perhaps more illuminating to check out the finite version. Let $X\in\mathbb R^N$ be a random Gaussian vector of dimension $N$, $0$ mean, and covariance matrix $C$ ie $\langle X_i X_j\rangle = C_{ij}$ with $C$ positive definite. All the statistics is captured by $C$. Let $J \in\mathbb R$, you can calculate: $$ \langle e^{J^TX}\rangle = \int e^{J^T X}\frac{e^{-X^TC^{-1}X/2}}{\sqrt{(2\pi)^N\det(C)}}d^N X \\ = e^{\frac{1}{2}J^TCJ}\int \frac{e^{-\frac{1}{2}(X-CJ)^TC^{-1}(X-CJ)}}{\sqrt{(2\pi)^N\det(C)}}d^N X \\ = e^{\frac{1}{2}J^TCJ} $$ (completed the square and integration variable translation)
To see the analogy better, you can rewrite: $$ \langle e^{\phi(x_1)}...e^{\phi(x_n)} \rangle = \langle e^{\int dx J(x)\phi(x)} \rangle $$ so you just need to formally set the dimension to the continuum and replace sums in the the dot product by integrals.
The importance of this is that generally you write $\mathcal Z(J) = \langle e^{\int dx J(x)\phi(x)} \rangle$ the generating functional as it gives you by successive differentiation at $0$ the Green's functions which correspond to correlators.
The characteristic of Gaussians is Wick's theorem, which says that higher order correlators are related to the two point correlator by a specific formula. This is how you can check whether a process is Gaussian or not.
Hope tis helps.
