Random walk on $\mathbf{Z}_d$ with Jacobi $\theta$ transition probabilities

In the context of a finite-dimensional quantum mechanical problem, I was led to study the random walk on $$\mathbf{Z}_d$$ (i.e the integers modulo $$d$$), $$d$$ odd with transition probabilities given by:

$$p_{kl} = \sqrt{\frac{\xi^2}{d}}\frac{1}{\theta_3\left(0, \frac{i}{d\xi^2}\right)}\theta_3\left(\frac{l - k}{d}, \frac{i\xi^2}{d}\right)$$

where $$\theta_3$$ is one of Jacobi's theta functions:

$$\theta_3(z, \tau) = \sum_{n \in \mathbf{Z}}\exp\left(i\pi\tau n^2 + 2\pi inz\right) \qquad \Im(\tau) > 0, z \in \mathbf{C}$$

which when sampled on a period enjoys nice properties under discrete Fourier transform:

$$\theta_3\left(z + \frac{k}{d}, \frac{i\xi^2}{d}\right) = \frac{1}{\sqrt{d\xi^2}}\sum_{-\frac{d - 1}{2} \leq j \leq \frac{d - 1}{2}}\theta_3\left(\frac{iz}{\xi^2} - \frac{j}{d}, \frac{i}{d\xi^2}\right)\exp\left(-\frac{\pi d}{\xi^2}z^2 + \frac{2\pi ijk}{d}\right)$$

$$\theta_3\left(\frac{iz}{\xi^2} - \frac{k}{d}, \frac{i}{d\xi^2}\right) = \sqrt{\frac{\xi^2}{d}}\sum_{-\frac{d - 1}{2} \leq j \leq \frac{d - 1}{2}}\theta_3\left(z + \frac{j}{d}, \frac{i\xi^2}{d}\right)\exp\left(\frac{\pi d}{\xi^2}z^2 - \frac{2\pi ijk}{d}\right)$$

Given a random walk consisting of $$M$$ such steps, I am interested in the number of times one reaches $$\frac{d - 1}{2}$$. More precisely, I would like to compute the probability that this number is even minus the probability that it is odd.

Heuristically, the answer is evident in two limiting cases. On the one hand, for "small" $$\xi$$ (given a fixed $$M$$), the distribution for one step is very narrow and the random walk will be unlikely to reach $$\frac{d - 1}{2}$$ at all and I expect the result to be close to $$1$$ up to an error decaying exponentially in $$d$$. On the other hand, in the limit $$\xi \to \infty$$ (more precisely for large enough $$\xi$$ given a fixed $$M$$), the distribution for each step will look like a uniform distribution and it is easy to find that the sought result should be equivalent to $$\left(1 - \frac{2}{d}\right)^M$$.

Now, I am wondering what precise statements can be made in the general setting. I would be very surprised if there existed a thorough answer but at the same time, it would be great to understand whether or not the two aforementioned limiting cases are really everything one can do...

• $\mathbf{Z}_d$ is a notation for integers modulo $d$, or for $d$-adic numbers? – YCor Feb 4 at 1:06
• For integers modulo $d$, thanks for the remark! – IchKenneDeinenNamen Feb 4 at 9:39