# Sample discrete multivariate normal

What is an efficient way of sampling from a discrete multivariate normal distribution with pdf

$$p(z) = \frac{1}{Z} e ^ {-\frac{1}{2}(z-\mu)^\top\Sigma^{-1}(z-\mu)}$$ such that $z \in \mathbb{Z}_k^n$? Here $\mathbb{Z}_k = \{z : -k \le z \le k, z\in\mathbb{Z}\}$.

I've contemplated rejection sampling based on drawing uniform samples from $\mathbb{Z}_k^n$, but this scales poorly with $n$.

I have found some extended discussions of sampling from 1D discrete Gaussians, but in my case I can not assume that $\Sigma$ is the identity, so I'm not sure how that is helpful to me.