The probability distribution of the area bounded by a Gaussian random contour

Let $\rho_i=(\rho_i^1,\rho_i^2)$, $i=1,\dots,N$, be two-dimensional Gaussian random variables with $\mathbf M\rho_i=(r\cos\varphi_i,r\sin\varphi_i)$, $\operatorname{cov}(\rho_i^\alpha,\rho_i^\beta)=\delta_{\alpha\beta}\biggl(4r^2\sin^2\frac{\varphi_i-\varphi_j}2\biggr)$, where $r$ is constant, $0\le\varphi_1<\dots<\varphi_N<2\pi$, and $g$ is a real function. Let $S_N$ be the area bounded by the broken line passing through the points $\rho_1,\dots,\rho_N$. In the paper, the distribution of $S=\lim\limits_{N\to\infty}S_N$ is studied.