As a follow up to 

http://mathoverflow.net/questions/234148/how-to-compute-sum-x-in-mathbbzn-e-xtmx-efficiently

Suppose that $M$ is a large Toeplitz matrix. With a suitable scaling $K^{-n}$ for some $K$, what will the Riemann theta function converge to as $n\to \infty$ ?
Would this somehow be simpler than the finite dimension Riemann theta function evaluation? Is there any progress?

I also have another, much less important for me, question.
If one changes a bit, so that $x=[\exp(i\alpha_1) \;\ldots \; \exp(i\alpha_n)]^T$ is there any hope to evaluate

$$\int\cdots \int \exp\left(-x^T Mx+2\mathrm{Re}(x^Ty)\right)\, d\alpha_1 \cdots d\alpha_n$$

??
(essentially a continuous version but over separate unit circles instead of over square lattices)