Let L be a linear differential operator with eigenfuctions $\phi\_{k}$ then for $a\_{n}\sim N(0,1)$ consider


$$h(z):=\sum a\_{n}\phi\_{k}(z).$$


Is there a general theory for such sums? For $L=\Delta$, h is called the GFF and has some interesting properties such as the GFF Markov property and its circular average being a Brownian motion (see [Berestyki survey](http://www.statslab.cam.ac.uk/~beresty/Articles/oxford.pdf)).



Reference: random sum of eigenfunctions $\sum a_{n}\phi_{k}(z)$, $a_{n}\sim N(0,1)$
Check out (on arxiv.org, if you like) work by Nazarov and Sodin and Sarnak and Wigman - they study many aspects of this sort of thing.


