As a follow up to

How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ 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)