Cutoff for Contingency Table Random Walks

Abstract: We study the Diaconis-Gangolli random walk on $n \times n$ contingency tables and its analog on $1 \times n$ contingency tables, both 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 $n \times n$ case, we establish cutoff for the random walk at time $\dfrac{n^2 q^2 \log(n)}{8 \pi^2}$ when $q \gg n^2$. We also show that a general class of random walks on $(\mathbb{Z}/q\mathbb{Z})^n$ with a marginal incremental variance $\dfrac{\sigma^2}{n}$ (when mapped to $( \mathbb{Z} \cap [-q/2, q/2))^n$) has cutoff at time $\dfrac{nq^2 \log(n)}{4\pi^2 \sigma^2}$.