We use the correlation matrix of the generating distribution to determine the mixing time for random walks on the torus $(\mathbb{ Z}/q\mathbb{ Z})^n$. We present our method in the context of the Diaconis-Gangolli random walk on both the $1 \times n$ and $m \times n$ contingency tables over $\mathbb{ Z}/q\mathbb{ Z}$. In the $1 \times n$ case, we prove that the random walk exhibits cutoff at time $\dfrac{ n q^2 \log(n)}{ 8 \pi^2}$ when $q \gg n$; in the $m \times n$ case, where $m, n$ are of the same order, we establish cutoff for the random walk at time $\dfrac{ mn q^2 \log(mn)}{ 16 \pi^2}$ when $q \gg n^2$. Our method reveals that a general class of random walks on the torus $(\mathbb{ Z}/q\mathbb{ Z})^n$ has cutoff. If each coordinate of the lifted random walk onto $\mathbb{ Z}^n$ has variance $\sigma^2/n$ in each jump, and the between-coordinate correlations are sufficiently low, then cutoff occurs at time $\dfrac{ nq^2 \log(n)}{ 4\pi^2 \sigma^2}$.
Categories: math.PR