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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.16203 |
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Table of Contents:
- 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 $\dfrac{n q^2 \log(n)}{8 π^2}$ 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 $\dfrac{mn q^2 \log(mn)}{16 π^2}$ 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 $σ^2/n$ in each jump, and the between-coordinate correlations are sufficiently low, then cutoff occurs at time $\dfrac{nq^2 \log(n)}{4π^2 σ^2}$.